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I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that

$\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)$

whenever ${A,B}$ are compact non-empty subsets of a compact connected additive group ${G}$ with probability Haar measure ${\mu}$.  (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

$\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)$

The connectedness of ${G}$ is essential, otherwise one could form counterexamples involving proper subgroups of ${G}$ of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

$\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B)$

$\displaystyle \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)$

which in turn easily follows from the identity ${(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)}$ and the inclusion ${(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}$, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking ${G}$ to be the unit circle ${G = {\bf R}/{\bf Z}}$ and ${A,B}$ to be arcs in that circle (obeying (2)). A bit more generally, if ${G}$ is an arbitrary connected compact abelian group and ${\xi: G \rightarrow {\bf R}/{\bf Z}}$ is a non-trivial character (i.e., a continuous homomorphism), then ${\xi}$ must be surjective (as ${{\bf R}/{\bf Z}}$ has no non-trivial connected subgroups), and one can take ${A = \xi^{-1}(I)}$ and ${B = \xi^{-1}(J)}$ for some arcs ${I,J}$ in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then ${A,B}$ must be close (in measure) to an example of the above form ${A = \xi^{-1}(I), B = \xi^{-1}(J)}$. Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset ${A+B}$ is replaced by the partial sumset ${A +_\varepsilon B}$ consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call ${(A,B)}$ a critical pair if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in ${A,B}$, and invariant with respect to translation of either ${A}$ or ${B}$. Furthermore, from the submodularity inequality (3), one can show that if ${(A_1,B)}$ and ${(A_2,B)}$ are critical pairs (with ${\mu(A_1 \cap A_2)}$ and ${1 - \mu(A_1 \cup A_2) - \mu(B)}$ positive), then ${(A_1 \cap A_2,B)}$ and ${(A_1 \cup A_2, B)}$ are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when ${A,B}$ come from arcs of a circle.) Similarly, from associativity ${(A+B)+C = A+(B+C)}$, one can show that if ${(A,B)}$ and ${(A+B,C)}$ are critical pairs, then so are ${(B,C)}$ and ${(A,B+C)}$.

One can combine these closure properties to obtain further ones. For instance, suppose ${A,B}$ is such that ${\mu(A+B) 0}$. Then (cheating a little bit), one can show that ${(A+B,C)}$ is also a critical pair, basically because ${A+B}$ is the union of the ${A+b}$, ${b \in B}$, the ${(A+b,C)}$ are all critical pairs, and the ${A+b}$ all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate ${A+B}$ by the union of finitely many of the ${A+b}$, then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair ${(A,B)}$ and end up with a small set ${C}$ such that ${(A,C)}$ and ${(kC,C)}$ are also critical pairs for all ${1 \leq k \leq 10^4}$ (say), where ${kC}$ is the ${k}$-fold sumset of ${C}$. (Intuitively, if ${A,B}$ are thought of as secretly coming from the pullback of arcs ${I,J}$ by some character ${\xi}$, then ${C}$ should be the pullback of a much shorter arc by the same character.) In particular, ${C}$ exhibits linear growth, in that ${\mu(kC) = k\mu(C)}$ for all ${1 \leq k \leq 10^4}$. One can now use standard technology from inverse sumset theory to show first that ${C}$ has a very large Fourier coefficient (and thus is biased with respect to some character ${\xi}$), and secondly that ${C}$ is in fact almost of the form ${C = \xi^{-1}(K)}$ for some arc ${K}$, from which it is not difficult to conclude similar statements for ${A}$ and ${B}$ and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman.  Thanks to John Griesmer for pointing out the error.]

Let ${\lambda: {\bf N} \rightarrow \{-1,1\}}$ be the Liouville function, thus ${\lambda(n)}$ is defined to equal ${+1}$ when ${n}$ is the product of an even number of primes, and ${-1}$ when ${n}$ is the product of an odd number of primes. The Chowla conjecture asserts that ${\lambda}$ has the statistics of a random sign pattern, in the sense that

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0 \ \ \ \ \ (1)$

for all ${k \geq 1}$ and all distinct natural numbers ${h_1,\dots,h_k}$, where we use the averaging notation

$\displaystyle \mathbb{E}_{n \leq N} f(n) := \frac{1}{N} \sum_{n \leq N} f(n).$

For ${k=1}$, this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any ${k \geq 2}$.

In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} \lambda(n+h_1) \dots \lambda(n+h_k) = 0 \ \ \ \ \ (2)$

of the conjecture, where we use the logarithmic averaging notation

$\displaystyle \mathbb{E}_{n \leq N}^{\log} f(n) := \frac{\sum_{n \leq N} \frac{f(n)}{n}}{\sum_{n \leq N} \frac{1}{n}}.$

Using the summation by parts (or telescoping series) identity

$\displaystyle \sum_{n \leq N} \frac{f(n)}{n} = \sum_{M < N} \frac{1}{M(M+1)} (\sum_{n \leq M} f(n)) + \frac{1}{N} \sum_{n \leq N} f(n) \ \ \ \ \ (3)$

it is not difficult to show that the Chowla conjecture (1) for a given ${k,h_1,\dots,h_k}$ implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for ${k=1}$, we have already mentioned that the Chowla conjecture

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N} \lambda(n) = 0$

is equivalent to the prime number theorem; but the logarithmically averaged analogue

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}^{\log}_{n \leq N} \lambda(n) = 0$

is significantly easier to show (a proof with the Liouville function ${\lambda}$ replaced by the closely related Möbius function ${\mu}$ is given in this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for ${k=2}$, and in this recent paper with Joni Teravainen, we proved the conjecture for all odd ${k}$ (with a different proof also given here).

In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:

Theorem 1 Assume that the logarithmically averaged Chowla conjecture (2) is true for all ${k}$. Then there exists a sequence ${N_i}$ going to infinity such that the Chowla conjecture (1) is true for all ${k}$ along that sequence, that is to say

$\displaystyle \lim_{N_i \rightarrow \infty} \mathbb{E}_{n \leq N_i} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

for all ${k}$ and all distinct ${h_1,\dots,h_k}$.

This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.

On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:

Theorem 2 Let ${k}$ be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for ${2k}$. Then there exists a set ${{\mathcal N}}$ of natural numbers of logarithmic density ${1}$ (that is, ${\lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} 1_{n \in {\mathcal N}} = 1}$) such that

$\displaystyle \lim_{N \rightarrow \infty: N \in {\mathcal N}} \mathbb{E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

for any distinct ${h_1,\dots,h_k}$.

It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture (${k=2}$ and odd ${k}$) are currently insufficient to use Theorem 2 for any purpose other than to reprove what is already known to be true from the prime number theorem. (Indeed, the even cases of Chowla, in either logarithmically averaged or non-logarithmically averaged forms, seem to be far more powerful than the odd cases; see Remark 1.7 of this paper of myself and Teravainen for a related observation in this direction.)

We now sketch the proof of Theorem 2. For any distinct ${h_1,\dots,h_k}$, we take a large number ${H}$ and consider the limiting the second moment

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{n \leq N}^{\log} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2.$

We can expand this as

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{m,m' \leq H} \mathop{\bf E}_{n \leq N}^{\log} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)$

$\displaystyle \lambda(n+m'+h_1) \dots \lambda(n+m'+h_k).$

If all the ${m+h_1,\dots,m+h_k,m'+h_1,\dots,m'+h_k}$ are distinct, the hypothesis (2) tells us that the inner averages goes to zero as ${N \rightarrow \infty}$. The remaining averages are ${O(1)}$, and there are ${O( k^2 )}$ of these averages. We conclude that

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{n \leq N}^{\log} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k^2 / H.$

By Markov’s inequality (and (3)), we conclude that for any fixed ${h_1,\dots,h_k, H}$, there exists a set ${{\mathcal N}_{h_1,\dots,h_k,H}}$ of upper logarithmic density at least ${1-k/H^{1/2}}$, thus

$\displaystyle \limsup_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} 1_{n \in {\mathcal N}_{h_1,\dots,h_k,H}} \geq 1 - k/H^{1/2}$

such that

$\displaystyle \mathop{\bf E}_{n \leq N} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k / H^{1/2}.$

By deleting at most finitely many elements, we may assume that ${{\mathcal N}_{h_1,\dots,h_k,H}}$ consists only of elements of size at least ${H^2}$ (say).

For any ${H_0}$, if we let ${{\mathcal N}_{h_1,\dots,h_k, \geq H_0}}$ be the union of ${{\mathcal N}_{h_1,\dots,h_k, H}}$ for ${H \geq H_0}$, then ${{\mathcal N}_{h_1,\dots,h_k, \geq H_0}}$ has logarithmic density ${1}$. By a diagonalisation argument (using the fact that the set of tuples ${(h_1,\dots,h_k)}$ is countable), we can then find a set ${{\mathcal N}}$ of natural numbers of logarithmic density ${1}$, such that for every ${h_1,\dots,h_k,H_0}$, every sufficiently large element of ${{\mathcal N}}$ lies in ${{\mathcal N}_{h_1,\dots,h_k,\geq H_0}}$. Thus for every sufficiently large ${N}$ in ${{\mathcal N}}$, one has

$\displaystyle \mathop{\bf E}_{n \leq N} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k / H^{1/2}.$

for some ${H \geq H_0}$ with ${N \geq H^2}$. By Cauchy-Schwarz, this implies that

$\displaystyle \mathop{\bf E}_{n \leq N} \mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k) \ll k^{1/2} / H^{1/4};$

interchanging the sums and using ${N \geq H^2}$ and ${H \geq H_0}$, this implies that

$\displaystyle \mathop{\bf E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) \ll k^{1/2} / H^{1/4} \leq k^{1/2} / H_0^{1/4}.$

We conclude on taking ${H_0}$ to infinity that

$\displaystyle \lim_{N \rightarrow \infty; N \in {\mathcal N}} \mathop{\bf E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

as required.

Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions.  Roth’s theorem is the special case when one considers arithmetic progressions of length three.  Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory.  However, the original proof by Endre Szemerédi, while extremely intricate, was purely combinatorial (and in particular “elementary”) and almost entirely self-contained, except for an invocation of the van der Waerden theorem.  It is also notable for introducing a prototype of what is now known as the Szemerédi regularity lemma.

Back in 2005, I rewrote Szemerédi’s original proof in order to understand it better, however my rewrite ended up being about the same length as the original argument and was probably only usable to myself.  In 2012, after Szemerédi was awarded the Abel prize, I revisited this argument with the intention to try to write up a more readable version of the proof, but ended up just presenting some ingredients of the argument in a blog post, rather than try to rewrite the whole thing.  In that post, I suspected that the cleanest way to write up the argument would be through the language of nonstandard analysis (perhaps in an iterated hyperextension that could handle various hierarchies of infinitesimals), but was unable to actually achieve any substantial simplifications by passing to the nonstandard world.

A few weeks ago, I participated in a week-long workshop at the American Institute of Mathematics on “Nonstandard methods in combinatorial number theory”, and spent some time in a working group with Shabnam Akhtari, Irfam Alam, Renling Jin, Steven Leth, Karl Mahlburg, Paul Potgieter, and Henry Towsner to try to obtain a manageable nonstandard version of Szemerédi’s original proof.  We didn’t end up being able to do so – in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument – but we did at least clarify the existing standard argument, to the point that I was able to go back to my original rewrite of the proof and present it in a more civilised form, which I am now uploading here as an unpublished preprint.   There are now a number of simplifications to the proof.  Firstly, one no longer needs the full strength of the regularity lemma; only the simpler “weak” regularity lemma of Frieze and Kannan is required.  Secondly, the proof has been “factored” into a number of stand-alone propositions of independent interest, in particular involving just (families of) one-dimensional arithmetic progressions rather than the complicated-looking multidimensional arithmetic progressions that occur so frequently in the original argument of Szemerédi.  Finally, the delicate manipulations of densities and epsilons via double counting arguments in Szemerédi’s original paper have been abstracted into a certain key property of families of arithmetic progressions that I call the “double counting property”.

The factoring mentioned above is particularly simple in the case of proving Roth’s theorem, which is now presented separately in the above writeup.  Roth’s theorem seeks to locate a length three progression ${(P(1),P(2),P(3)) = (a, a+r, a+2r)}$ in which all three elements lie in a single set.  This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements ${P(1), P(2)}$ of the progression lie in a good set (and some other properties of the family are also required).  This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.

More specifically, Roth’s theorem is now deduced from

Theorem 1.5.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ and ${P_l(2)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(3)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation in this theorem is depicted by the following diagram, in which elements of $A$ are in blue and elements of $S$ are in grey:

Theorem 1.5 is deduced in turn from the following easier variant:

Theorem 1.6.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(2)}$ and ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(2)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation here is described by the figure below.

Theorem 1.6 is easy to prove.  To derive Theorem 1.5 from Theorem 1.6, or to derive Roth’s theorem from Theorem 1.5, one uses double counting arguments, van der Waerden’s theorem, and the weak regularity lemma, largely as described in this previous blog post; see the writeup for the full details.  (I would be interested in seeing a shorter proof of Theorem 1.5 though that did not go through these arguments, and did not use the more powerful theorems of  Roth or Szemerédi.)

Fix a non-negative integer ${k}$. Define an (weak) integer partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ of non-increasing non-negative integers ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$. (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the more usual notion of “strong” partitions.) To each such partition ${\lambda}$, one can associate a Young diagram consisting of ${k}$ left-justified rows of boxes, with the ${i^{th}}$ row containing ${\lambda_i}$ boxes. A semi-standard Young tableau (or Young tableau for short) ${T}$ of shape ${\lambda}$ is a filling of these boxes by integers in ${\{1,\dots,k\}}$ that is weakly increasing along rows (moving rightwards) and strictly increasing along columns (moving downwards). The collection of such tableaux will be denoted ${{\mathcal T}_\lambda}$. The weight ${|T|}$ of a tableau ${T}$ is the tuple ${(n_1,\dots,n_k)}$, where ${n_i}$ is the number of occurrences of the integer ${i}$ in the tableau. For instance, if ${k=3}$ and ${\lambda = (6,4,2)}$, an example of a Young tableau of shape ${\lambda}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 2 & 3 & 3 \\ \cline{1-6} 2 & 2 & 2 &3\\ \cline{1-4} 3 & 3\\ \cline{1-2} \end{tabular}$

The weight here would be ${|T| = (3,4,5)}$.

To each partition ${\lambda}$ one can associate the Schur polynomial ${s_\lambda(u_1,\dots,u_k)}$ on ${k}$ variables ${u = (u_1,\dots,u_k)}$, which we will define as

$\displaystyle s_\lambda(u) := \sum_{T \in {\mathcal T}_\lambda} u^{|T|}$

using the multinomial convention

$\displaystyle (u_1,\dots,u_k)^{(n_1,\dots,n_k)} := u_1^{n_1} \dots u_k^{n_k}.$

Thus for instance the Young tableau ${T}$ given above would contribute a term ${u_1^3 u_2^4 u_3^5}$ to the Schur polynomial ${s_{(6,4,2)}(u_1,u_2,u_3)}$. In the case of partitions of the form ${(n,0,\dots,0)}$, the Schur polynomial ${s_{(n,0,\dots,0)}}$ is just the complete homogeneous symmetric polynomial ${h_n}$ of degree ${n}$ on ${k}$ variables:

$\displaystyle s_{(n,0,\dots,0)}(u_1,\dots,u_k) := \sum_{n_1,\dots,n_k \geq 0: n_1+\dots+n_k = n} u_1^{n_1} \dots u_k^{n_k},$

thus for instance

$\displaystyle s_{(3,0)}(u_1,u_2) = u_1^3 + u_1^2 u_2 + u_1 u_2^2 + u_2^3.$

Schur polyomials are ubiquitous in the algebraic combinatorics of “type ${A}$ objects” such as the symmetric group ${S_k}$, the general linear group ${GL_k}$, or the unitary group ${U_k}$. For instance, one can view ${s_\lambda}$ as the character of an irreducible polynomial representation of ${GL_k({\bf C})}$ associated with the partition ${\lambda}$. However, we will not focus on these interpretations of Schur polynomials in this post.

This definition of Schur polynomials allows for a way to describe the polynomials recursively. If ${k > 1}$ and ${T}$ is a Young tableau of shape ${\lambda = (\lambda_1,\dots,\lambda_k)}$, taking values in ${\{1,\dots,k\}}$, one can form a sub-tableau ${T'}$ of some shape ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ by removing all the appearances of ${k}$ (which, among other things, necessarily deletes the ${k^{th}}$ row). For instance, with ${T}$ as in the previous example, the sub-tableau ${T'}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|} \hline 1 & 1 & 1 & 2 \\ \cline{1-4} 2 & 2 & 2 \\ \cline{1-3} \end{tabular}$

and the reduced partition ${\lambda'}$ in this case is ${(4,3)}$. As Young tableaux are required to be strictly increasing down columns, we can see that the reduced partition ${\lambda'}$ must intersperse the original partition ${\lambda}$ in the sense that

$\displaystyle \lambda_{i+1} \leq \lambda'_i \leq \lambda_i \ \ \ \ \ (1)$

for all ${1 \leq i \leq k-1}$; we denote this interspersion relation as ${\lambda' \prec \lambda}$ (though we caution that this is not intended to be a partial ordering). In the converse direction, if ${\lambda' \prec \lambda}$ and ${T'}$ is a Young tableau with shape ${\lambda'}$ with entries in ${\{1,\dots,k-1\}}$, one can form a Young tableau ${T}$ with shape ${\lambda}$ and entries in ${\{1,\dots,k\}}$ by appending to ${T'}$ an entry of ${k}$ in all the boxes that appear in the ${\lambda}$ shape but not the ${\lambda'}$ shape. This one-to-one correspondence leads to the recursion

$\displaystyle s_\lambda(u) = \sum_{\lambda' \prec \lambda} s_{\lambda'}(u') u_k^{|\lambda| - |\lambda'|} \ \ \ \ \ (2)$

where ${u = (u_1,\dots,u_k)}$, ${u' = (u_1,\dots,u_{k-1})}$, and the size ${|\lambda|}$ of a partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ is defined as ${|\lambda| := \lambda_1 + \dots + \lambda_k}$.

One can use this recursion (2) to prove some further standard identities for Schur polynomials, such as the determinant identity

$\displaystyle s_\lambda(u) V(u) = \det( u_i^{\lambda_j+k-j} )_{1 \leq i,j \leq k} \ \ \ \ \ (3)$

for ${u=(u_1,\dots,u_k)}$, where ${V(u)}$ denotes the Vandermonde determinant

$\displaystyle V(u) := \prod_{1 \leq i < j \leq k} (u_i - u_j), \ \ \ \ \ (4)$

or the Jacobi-Trudi identity

$\displaystyle s_\lambda(u) = \det( h_{\lambda_j - j + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (5)$

with the convention that ${h_d(u) = 0}$ if ${d}$ is negative. Thus for instance

$\displaystyle s_{(1,1,0,\dots,0)}(u) = h_1^2(u) - h_0(u) h_2(u) = \sum_{1 \leq i < j \leq k} u_i u_j.$

We review the (standard) derivation of these identities via (2) below the fold. Among other things, these identities show that the Schur polynomials are symmetric, which is not immediately obvious from their definition.

One can also iterate (2) to write

$\displaystyle s_\lambda(u) = \sum_{() = \lambda^0 \prec \lambda^1 \prec \dots \prec \lambda^k = \lambda} \prod_{j=1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (6)$

where the sum is over all tuples ${\lambda^1,\dots,\lambda^k}$, where each ${\lambda^j}$ is a partition of length ${j}$ that intersperses the next partition ${\lambda^{j+1}}$, with ${\lambda^k}$ set equal to ${\lambda}$. We will call such a tuple an integral Gelfand-Tsetlin pattern based at ${\lambda}$.

One can generalise (6) by introducing the skew Schur functions

$\displaystyle s_{\lambda/\mu}(u) := \sum_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \prod_{j=i+1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (7)$

for ${u = (u_{i+1},\dots,u_k)}$, whenever ${\lambda}$ is a partition of length ${k}$ and ${\mu}$ a partition of length ${i}$ for some ${0 \leq i \leq k}$, thus the Schur polynomial ${s_\lambda}$ is also the skew Schur polynomial ${s_{\lambda /()}}$ with ${i=0}$. (One could relabel the variables here to be something like ${(u_1,\dots,u_{k-i})}$ instead, but this labeling seems slightly more natural, particularly in view of identities such as (8) below.)

By construction, we have the decomposition

$\displaystyle s_{\lambda/\nu}(u_{i+1},\dots,u_k) = \sum_\mu s_{\mu/\nu}(u_{i+1},\dots,u_j) s_{\lambda/\mu}(u_{j+1},\dots,u_k) \ \ \ \ \ (8)$

whenever ${0 \leq i \leq j \leq k}$, and ${\nu, \mu, \lambda}$ are partitions of lengths ${i,j,k}$ respectively. This gives another recursive way to understand Schur polynomials and skew Schur polynomials. For instance, one can use it to establish the generalised Jacobi-Trudi identity

$\displaystyle s_{\lambda/\mu}(u) = \det( h_{\lambda_j - j - \mu_i + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (9)$

with the convention that ${\mu_i = 0}$ for ${i}$ larger than the length of ${\mu}$; we do this below the fold.

The Schur polynomials (and skew Schur polynomials) are “discretised” (or “quantised”) in the sense that their parameters ${\lambda, \mu}$ are required to be integer-valued, and their definition similarly involves summation over a discrete set. It turns out that there are “continuous” (or “classical”) analogues of these functions, in which the parameters ${\lambda,\mu}$ now take real values rather than integers, and are defined via integration rather than summation. One can view these continuous analogues as a “semiclassical limit” of their discrete counterparts, in a manner that can be made precise using the machinery of geometric quantisation, but we will not do so here.

The continuous analogues can be defined as follows. Define a real partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ where ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$ are now real numbers. We can define the relation ${\lambda' \prec \lambda}$ of interspersion between a length ${k-1}$ real partition ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ and a length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_{k})}$ precisely as before, by requiring that the inequalities (1) hold for all ${1 \leq i \leq k-1}$. We can then define the continuous Schur functions ${S_\lambda(x)}$ for ${x = (x_1,\dots,x_k) \in {\bf R}^k}$ recursively by defining

$\displaystyle S_{()}() = 1$

and

$\displaystyle S_\lambda(x) = \int_{\lambda' \prec \lambda} S_{\lambda'}(x') \exp( (|\lambda| - |\lambda'|) x_k ) \ \ \ \ \ (10)$

for ${k \geq 1}$ and ${\lambda}$ of length ${k}$, where ${x' := (x_1,\dots,x_{k-1})}$ and the integral is with respect to ${k-1}$-dimensional Lebesgue measure, and ${|\lambda| = \lambda_1 + \dots + \lambda_k}$ as before. Thus for instance

$\displaystyle S_{(\lambda_1)}(x_1) = \exp( \lambda_1 x_1 )$

and

$\displaystyle S_{(\lambda_1,\lambda_2)}(x_1,x_2) = \int_{\lambda_2}^{\lambda_1} \exp( \lambda'_1 x_1 + (\lambda_1+\lambda_2-\lambda'_1) x_2 )\ d\lambda'_1.$

More generally, we can define the continuous skew Schur functions ${S_{\lambda/\mu}(x)}$ for ${\lambda}$ of length ${k}$, ${\mu}$ of length ${j \leq k}$, and ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$ recursively by defining

$\displaystyle S_{\mu/\mu}() = 1$

and

$\displaystyle S_{\lambda/\mu}(x) = \int_{\lambda' \prec \lambda} S_{\lambda'/\mu}(x') \exp( (|\lambda| - |\lambda'|) x_k )$

for ${k > j}$. Thus for instance

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1,\mu_2)}(x_3) = 1_{\lambda_3 \leq \mu_2 \leq \lambda_2 \leq \mu_1 \leq \lambda_1} \exp( x_3 (\lambda_1+\lambda_2+\lambda_3 - \mu_1 - \mu_2 ))$

and

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1)}(x_2, x_3) = \int_{\lambda_2 \leq \lambda'_2 \leq \lambda_2, \mu_1} \int_{\mu_1, \lambda_2 \leq \lambda'_1 \leq \lambda_1}$

$\displaystyle \exp( x_2 (\lambda'_1+\lambda'_2 - \mu_1) + x_3 (\lambda_1+\lambda_2+\lambda_3 - \lambda'_1 - \lambda'_2))\ d\lambda'_1 d\lambda'_2.$

By expanding out the recursion, one obtains the analogue

$\displaystyle S_\lambda(x) = \int_{\lambda^1 \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^1 \dots d\lambda^{k-1},$

of (6), and more generally one has

$\displaystyle S_{\lambda/\mu}(x) = \int_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=i+1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^{i+1} \dots d\lambda^{k-1}.$

We will call the tuples ${(\lambda^1,\dots,\lambda^k)}$ in the first integral real Gelfand-Tsetlin patterns based at ${\lambda}$. The analogue of (8) is then

$\displaystyle S_{\lambda/\nu}(x_{i+1},\dots,x_k) = \int S_{\mu/\nu}(x_{i+1},\dots,x_j) S_{\lambda/\mu}(x_{j+1},\dots,x_k)\ d\mu$

where the integral is over all real partitions ${\mu}$ of length ${j}$, with Lebesgue measure.

By approximating various integrals by their Riemann sums, one can relate the continuous Schur functions to their discrete counterparts by the limiting formula

$\displaystyle N^{-k(k-1)/2} s_{\lfloor N \lambda \rfloor}( \exp[ x/N ] ) \rightarrow S_\lambda(x) \ \ \ \ \ (11)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ and any ${x = (x_1,\dots,x_k) \in {\bf R}^k}$, where

$\displaystyle \lfloor N \lambda \rfloor := ( \lfloor N \lambda_1 \rfloor, \dots, \lfloor N \lambda_k \rfloor )$

and

$\displaystyle \exp[x/N] := (\exp(x_1/N), \dots, \exp(x_k/N)).$

More generally, one has

$\displaystyle N^{j(j-1)/2-k(k-1)/2} s_{\lfloor N \lambda \rfloor / \lfloor N \mu \rfloor}( \exp[ x/N ] ) \rightarrow S_{\lambda/\mu}(x)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda}$, any length ${j}$ real partition ${\mu}$ with ${0 \leq j \leq k}$, and any ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$.

As a consequence of these limiting formulae, one expects all of the discrete identities above to have continuous counterparts. This is indeed the case; below the fold we shall prove the discrete and continuous identities in parallel. These are not new results by any means, but I was not able to locate a good place in the literature where they are explicitly written down, so I thought I would try to do so here (primarily for my own internal reference, but perhaps the calculations will be worthwhile to some others also).

Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for ${r_4(N)}$“, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This sequel has been delayed for over a decade for a number of reasons, but we have finally managed to write the arguments up to our satisfaction and submit it (to a special issue of Mathematika honouring the work of Klaus Roth).

For any natural number ${N}$, define ${r_4(N)}$ to be the largest cardinality of a subset ${A}$ of ${[N] = \{1,\dots,N\}}$ which does not contain any non-trivial arithmetic progressions ${a, a+r, a+2r, a+3r}$ of length four (where “non-trivial” means that ${r}$ is non-zero). Trivially we have ${r_4(N) \leq N}$. In 1969, Szemerédi showed that ${r_4(N) = o(N)}$. However, the decay rate that could be theoretically extracted from this argument (and from several subsequent proofs of this bound, including one by Roth) were quite poor. The first significant quantitative bound on this quantity was by Gowers, who showed that ${r_4(N) \ll N (\log \log N)^{-c}}$ for some absolute constant ${c>0}$. In the second paper in the above-mentioned series, we managed to improve this bound to ${r_4(N) \ll N \exp( - c \sqrt{\log \log N})}$. In this paper, we improve the bound further to ${r_4(N) \ll N (\log N)^{-c}}$, which seems to be the limit of the methods. (We remark that if we could take ${c}$ to be larger than one, this would imply the length four case of a well known conjecture of Erdös that any set of natural numbers whose sum of reciprocals diverges would contain arbitrarily long arithmetic progressions. Thanks to the work of Sanders and of Bloom, the corresponding case of the conjecture for length three conjectures is nearly settled, as it is known that for the analogous bound on ${r_3(N)}$ one can take any ${c}$ less than one.)

Most of the previous work on bounding ${r_4(N)}$ relied in some form or another on the density increment argument introduced by Roth back in 1953; roughly speaking, the idea is to show that if a dense subset ${A}$ of ${[N]}$ fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of ${[N]}$ in which ${A}$ has increased density. This was the basic method for instance underlying our previous bound ${r_4(N) \ll N \exp( - c \sqrt{\log \log N})}$, as well as a finite field analogue of the bound ${r_4(N) \ll N (\log N)^{-c}}$; however we encountered significant technical difficulties for several years in extending this argument to obtain the result of the current paper. Our method is instead based on “energy increment arguments”, and more specifically on establishing quantitative version of a Khintchine-type recurrence theorem, similar to the qualitative recurrence theorems established (in the ergodic theory context) by Bergelson-Host-Kra, and (in the current combinatorial context) by Ben Green and myself.

One way to phrase the latter recurrence theorem is as follows. Suppose that ${A \subset [N]}$ has density ${\delta}$. Then one would expect a “randomly” selected arithmetic progression ${{\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r}}$ in ${[N]}$ (using the convention that random variables will be in boldface) to be contained in ${A}$ with probability about ${\delta^4}$. This is not true in general, however it was shown by Ben and myself that for any ${\eta>0}$, there was a set of shifts ${r \in [-N,N]}$ of cardinality ${\gg_{\delta,\eta} N}$, such that for any such ${r}$ one had

$\displaystyle {\bf P}( {\bf a}, {\bf a}+r, {\bf a}+2r, {\bf a}+3r \in A ) \geq \delta^4 - \eta$

if ${{\bf a}}$ was chosen uniformly at random from ${[N]}$. This easily implies that ${r_4(N) = o(N)}$, but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound ${\gg_{\delta,\eta} N}$ is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take ${N}$ to be extremely large compared to ${\delta,\eta}$ to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift ${r=0}$.

We do not know how to improve the lower bound on the set of shifts to the point where it can give bounds that are competitive with those in this paper. However, we can obtain better quantitative results if we permit ourselves to couple together the two parameters ${{\bf a}}$ and ${{\bf r}}$ of the length four progression. Namely, with ${A}$, ${\delta}$, ${\eta}$ as above, we are able to show that there exist random variables ${{\bf a}, {\bf r}}$, not necessarily independent, such that

$\displaystyle {\bf P}( {\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r} \in A ) \geq \delta^4 - \eta \ \ \ \ \ (1)$

and such that we have the non-degeneracy bound

$\displaystyle {\bf P}( {\bf r} = 0 ) \ll \exp( - \eta^{-O(1)} ) / N.$

This then easily implies the main theorem.

The energy increment method is then deployed to locate a good pair ${({\bf a}, {\bf r})}$ of random variables that will obey the above bounds. One can get some intuition on how to proceed here by considering some model cases. Firstly one can consider a “globally quadratically structured” case in which the indicator function ${1_A}$ “behaves like” a globally quadratic function such as ${F( \alpha n^2 )}$, for some irrational ${\alpha}$ and some smooth periodic function ${F: {\bf R}/{\bf Z} \rightarrow {\bf R}}$ of mean ${\delta}$. If one then takes ${{\bf a}, {\bf r}}$ to be uniformly distributed in ${[N]}$ and ${[-\varepsilon N, \varepsilon N]}$ respectively for some small ${\varepsilon>0}$, with no coupling between the two variables, then the left-hand side of (1) is approximately of the form

$\displaystyle \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F(x) F(y) F(z) F(w) \ \ \ \ \ (2)$

where the integral is with respect to the probability Haar measure, and the constraint ${x-3y+3z-w=0}$ ultimately arises from the algebraic constraint

$\displaystyle \alpha {\bf a}^2 - 3 \alpha ({\bf a}+{\bf r})^2 + 3 \alpha ({\bf a}+2{\bf r})^2 - \alpha ({\bf a}+3{\bf r})^2 = 0.$

However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least ${(\int_{{\bf R}/{\bf Z}} F)^4}$, which (morally at least) gives (1) in this case.

Due to the nature of the energy increment argument, it also becomes necessary to consider “locally quadratically structured” cases, in which ${[N]}$ is partitioned into some number of structured pieces ${B_c}$ (think of these as arithmetic progressions, or as “Bohr sets), and on each piece ${B_c}$, ${1_A}$ behaves like a locally quadratic function such as ${F_c( \alpha_c n^2 )}$, where ${\alpha_c}$ now varies with ${c}$, and the mean of ${F_c}$ will be approximately ${\delta}$ on the average after averaging in ${c}$ (weighted by the size of the pieces ${B_c}$). Now one should select ${{\bf a}}$ and ${{\bf r}}$ in the following coupled manner: first one chooses ${{\bf a}}$ uniformly from ${[N]}$, then one defines ${{\bf c}}$ to be the label ${c}$ such that ${{\bf a} \in B_c}$, and then selects ${{\bf r}}$ uniformly from a set ${B_{c,\varepsilon}}$ which is related to ${B_c}$ in much the same way that ${[-\varepsilon N, \varepsilon N]}$ is related to ${[N]}$. If one does this correctly, the analogue of (2) becomes

$\displaystyle {\bf E} \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F_{\mathbf c}(x) F_{\mathbf c}(y) F_{\mathbf c}(z) F_{\mathbf c}(w),$

and one can again use Cauchy-Schwarz and Fubini’s theorem to conclude.

The general case proceeds, very roughly, by an iterative argument. At each stage of the iteration, one has some sort of quadratic model of ${1_A}$ which involves a decomposition of ${[N]}$ into structured pieces ${B_c}$, and a quadratic approximation to ${1_A}$ on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm ${U^3}$ of the error is small enough) to model the count in (1) (for random variables ${{\bf a}, {\bf r}}$ determined by the above partition of ${[N]}$ into pieces ${B_c}$), and if the frequencies (such as ${\alpha_c}$) involved in the quadratic approximation are “high rank” or “linearly independent over the rationals” in a suitably quantitative sense, then some version of the above arguments can be made to work. If there are some unwanted linear dependencies in the frequencies, we can do some linear algebra to eliminate one of the frequencies (using some geometry of numbers to keep the quantitative bounds under control) and continue the iteration. If instead the approximation is too inaccurate, then the error will be large in a certain averaged local Gowers uniformity norm ${U^3}$. A significant fraction of the paper is then devoted to establishing a quantitative inverse theorem for that norm that concludes (with good bounds) that the error must then locally correlate with locally quadratic phases, which can be used to refine the quadratic approximation to ${1_A}$ in a manner that significantly increases its “energy” (basically an ${L^2}$ norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.

There are existing inverse theorems for ${U^3}$ type norms in the literature, going back to the pioneering work of Gowers mentioned previously, and relying on arithmetic combinatorics tools such as Freiman’s theorem and the Balog-Szemerédi-Gowers lemma, which are good for analysing the “${1\%}$-structured homomorphisms” that arise in Gowers’ argument. However, when we applied these methods to the local Gowers norms we obtained inferior quantitative results that were not strong enough for our application. Instead, we use arguments from a different paper of Gowers in which he tackled Szemerédi’s theorem for arbitrary length progressions. This method produces “${99\%}$-structured homomorphisms” associated to any function with large Gowers uniformity norm; however the catch is that such homomorphisms are initially supported only on a sparse unstructured set, rather than a structured set such as a Bohr set. To proceed further, one first has to locate inside the sparse unstructured set a sparse pseudorandom subset of a Bohr set, and then use “error-correction” type methods (such as “majority-vote” based algorithms) to locally upgrade this ${99\%}$-structured homomorphism on pseudorandom subsets of Bohr sets to a ${100\%}$-structured homomorphism on the entirety of a Bohr set. It is then possible to use some “approximate cohomology” tools to “integrate” these homomorphisms (and discern a key “local symmetry” property of these homomorphisms) to locate the desired local quadratic structure (in much the same fashion that a ${1}$-form on ${{\bf R}^n}$ that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the ${1}$-form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.

How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because any four objects ${a,b,c,d}$ can be turned into a group ${\{a,b,c,d\}}$ by designating one of the four objects, say ${a}$, to be the group identity, and imposing a suitable multiplication table (and inversion law) on the four elements in a manner that obeys the usual group axioms. Since all sets are themselves objects, the class of four-element groups is thus at least as large as the class of all sets, which by Russell’s paradox is known not to itself be a set (assuming the usual ZFC axioms of set theory).

A much better question is to ask how many groups of order four there are up to isomorphism, counting each isomorphism class of groups exactly once. Now, as one learns in undergraduate group theory classes, the answer is just “two”: the cyclic group ${C_4}$ and the Klein four-group ${C_2 \times C_2}$.

More generally, given a class of objects ${X}$ and some equivalence relation ${\sim}$ on ${X}$ (which one should interpret as describing the property of two objects in ${X}$ being “isomorphic”), one can consider the number ${|X / \sim|}$ of objects in ${X}$ “up to isomorphism”, which is simply the cardinality of the collection ${X/\sim}$ of equivalence classes ${[x]:=\{y\in X:x \sim {}y \}}$ of ${X}$. In the case where ${X}$ is finite, one can express this cardinality by the formula

$\displaystyle |X/\sim| = \sum_{x \in X} \frac{1}{|[x]|}, \ \ \ \ \ (1)$

thus one counts elements in ${X}$, weighted by the reciprocal of the number of objects they are isomorphic to.

Example 1 Let ${X}$ be the five-element set ${\{-2,-1,0,1,2\}}$ of integers between ${-2}$ and ${2}$. Let us say that two elements ${x, y}$ of ${X}$ are isomorphic if they have the same magnitude: ${x \sim y \iff |x| = |y|}$. Then the quotient space ${X/\sim}$ consists of just three equivalence classes: ${\{-2,2\} = [2] = [-2]}$, ${\{-1,1\} = [-1] = [1]}$, and ${\{0\} = [0]}$. Thus there are three objects in ${X}$ up to isomorphism, and the identity (1) is then just

$\displaystyle 3 = \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2}.$

Thus, to count elements in ${X}$ up to equivalence, the elements ${-2,-1,1,2}$ are given a weight of ${1/2}$ because they are each isomorphic to two elements in ${X}$, while the element ${0}$ is given a weight of ${1}$ because it is isomorphic to just one element in ${X}$ (namely, itself).

Given a finite probability set ${X}$, there is also a natural probability distribution on ${X}$, namely the uniform distribution, according to which a random variable ${\mathbf{x} \in X}$ is set equal to any given element ${x}$ of ${X}$ with probability ${\frac{1}{|X|}}$:

$\displaystyle {\bf P}( \mathbf{x} = x ) = \frac{1}{|X|}.$

Given a notion ${\sim}$ of isomorphism on ${X}$, one can then define the random equivalence class ${[\mathbf{x}] \in X/\sim}$ that the random element ${\mathbf{x}}$ belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if ${\mathbf{x}}$ was drawn uniformly from ${X}$, the equivalence class ${[\mathbf{x}]}$ need not be uniformly distributed in ${X/\sim}$. For instance, in the above example, if ${\mathbf{x}}$ was drawn uniformly from ${\{-2,-1,0,1,2\}}$, then the equivalence class ${[\mathbf{x}]}$ will not be uniformly distributed in the three-element space ${X/\sim}$, because it has a ${2/5}$ probability of being equal to the class ${\{-2,2\}}$ or to the class ${\{-1,1\}}$, and only a ${1/5}$ probability of being equal to the class ${\{0\}}$.

However, it is possible to remove this bias by changing the weighting in (1), and thus changing the notion of what cardinality means. To do this, we generalise the previous situation. Instead of considering sets ${X}$ with an equivalence relation ${\sim}$ to capture the notion of isomorphism, we instead consider groupoids, which are sets ${X}$ in which there are allowed to be multiple isomorphisms between elements in ${X}$ (and in particular, there are allowed to be multiple automorphisms from an element to itself). More precisely:

Definition 2 A groupoid is a set (or proper class) ${X}$, together with a (possibly empty) collection ${\mathrm{Iso}(x \rightarrow y)}$ of “isomorphisms” between any pair ${x,y}$ of elements of ${X}$, and a composition map ${f, g \mapsto g \circ f}$ from isomorphisms ${f \in \mathrm{Iso}(x \rightarrow y)}$, ${g \in \mathrm{Iso}(y \rightarrow z)}$ to isomorphisms in ${\mathrm{Iso}(x \rightarrow z)}$ for every ${x,y,z \in X}$, obeying the following group-like axioms:

• (Identity) For every ${x \in X}$, there is an identity isomorphism ${\mathrm{id}_x \in \mathrm{Iso}(x \rightarrow x)}$, such that ${f \circ \mathrm{id}_x = \mathrm{id}_y \circ f = f}$ for all ${f \in \mathrm{Iso}(x \rightarrow y)}$ and ${x,y \in X}$.
• (Associativity) If ${f \in \mathrm{Iso}(x \rightarrow y)}$, ${g \in \mathrm{Iso}(y \rightarrow z)}$, and ${h \in \mathrm{Iso}(z \rightarrow w)}$ for some ${x,y,z,w \in X}$, then ${h \circ (g \circ f) = (h \circ g) \circ f}$.
• (Inverse) If ${f \in \mathrm{Iso}(x \rightarrow y)}$ for some ${x,y \in X}$, then there exists an inverse isomorphism ${f^{-1} \in \mathrm{Iso}(y \rightarrow x)}$ such that ${f \circ f^{-1} = \mathrm{id}_y}$ and ${f^{-1} \circ f = \mathrm{id}_x}$.

We say that two elements ${x,y}$ of a groupoid are isomorphic, and write ${x \sim y}$, if there is at least one isomorphism from ${x}$ to ${y}$.

Example 3 Any category gives a groupoid by taking ${X}$ to be the set (or class) of objects, and ${\mathrm{Iso}(x \rightarrow y)}$ to be the collection of invertible morphisms from ${x}$ to ${y}$. For instance, in the category ${\mathbf{Set}}$ of sets, ${\mathrm{Iso}(x \rightarrow y)}$ would be the collection of bijections from ${x}$ to ${y}$; in the category ${\mathbf{Vec}/k}$ of linear vector spaces over some given base field ${k}$, ${\mathrm{Iso}(x \rightarrow y)}$ would be the collection of invertible linear transformations from ${x}$ to ${y}$; and so forth.

Every set ${X}$ equipped with an equivalence relation ${\sim}$ can be turned into a groupoid by assigning precisely one isomorphism ${\iota_{x \rightarrow y}}$ from ${x}$ to ${y}$ for any pair ${x,y \in X}$ with ${x \sim y}$, and no isomorphisms from ${x}$ to ${y}$ when ${x \not \sim y}$, with the groupoid operations of identity, composition, and inverse defined in the only way possible consistent with the axioms. We will call this the simply connected groupoid associated with this equivalence relation. For instance, with ${X = \{-2,-1,0,1,2\}}$ as above, if we turn ${X}$ into a simply connected groupoid, there will be precisely one isomorphism from ${2}$ to ${-2}$, and also precisely one isomorphism from ${2}$ to ${2}$, but no isomorphisms from ${2}$ to ${-1}$, ${0}$, or ${1}$.

However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of ${X}$ to another. For instance, one can view ${X = \{-2,-1,0,1,2\}}$ as a space that is acted on by multiplication by the two-element group ${\{-1,+1\}}$. This gives rise to two types of isomorphisms, an identity isomorphism ${(+1)_x}$ from ${x}$ to ${x}$ for each ${x \in X}$, and a negation isomorphism ${(-1)_x}$ from ${x}$ to ${-x}$ for each ${x \in X}$; in particular, there are two automorphisms of ${0}$ (i.e., isomorphisms from ${0}$ to itself), namely ${(+1)_0}$ and ${(-1)_0}$, whereas the other four elements of ${X}$ only have a single automorphism (the identity isomorphism). One defines composition, identity, and inverse in this groupoid in the obvious fashion (using the group law of the two-element group ${\{-1,+1\}}$); for instance, we have ${(-1)_{-2} \circ (-1)_2 = (+1)_2}$.

For a finite multiply-connected groupoid, it turns out that the natural notion of “cardinality” (or as I prefer to call it, “cardinality up to isomorphism”) is given by the variant

$\displaystyle \sum_{x \in X} \frac{1}{|\{ f: f \in \mathrm{Iso}(x \rightarrow y) \hbox{ for some } y\}|}$

of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects ${y}$ isomorphic to ${x}$, but rather the number of isomorphisms from ${x}$ to other objects ${y}$. Grouping together all summands coming from a single equivalence class ${[x]}$ in ${X/\sim}$, we can also write this expression as

$\displaystyle \sum_{[x] \in X/\sim} \frac{1}{|\mathrm{Aut}(x)|} \ \ \ \ \ (2)$

where ${\mathrm{Aut}(x) := \mathrm{Iso}(x \rightarrow x)}$ is the automorphism group of ${x}$, that is to say the group of isomorphisms from ${x}$ to itself. (Note that if ${x,x'}$ belong to the same equivalence class ${[x]}$, then the two groups ${\mathrm{Aut}(x)}$ and ${\mathrm{Aut}(x')}$ will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.

For instance, if we take ${X}$ to be the simply connected groupoid on ${\{-2,-1,0,1,2\}}$, then the number of elements of ${X}$ up to isomorphism is

$\displaystyle \frac{1}{2} + \frac{1}{2} + 1 + \frac{1}{2} + \frac{1}{2} = 1 + 1 + 1 = 3$

exactly as before. If however we take the multiply connected groupoid on ${\{-2,-1,0,1,2\}}$, in which ${0}$ has two automorphisms, the number of elements of ${X}$ up to isomorphism is now the smaller quantity

$\displaystyle \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 1 + \frac{1}{2} + 1 = \frac{5}{2};$

the equivalence class ${[0]}$ is now counted with weight ${1/2}$ rather than ${1}$ due to the two automorphisms on ${0}$. Geometrically, one can think of this groupoid as being formed by taking the five-element set ${\{-2,-1,0,1,2\}}$, and “folding it in half” around the fixed point ${0}$, giving rise to two “full” quotient points ${[1], [2]}$ and one “half” point ${[0]}$. More generally, given a finite group ${G}$ acting on a finite set ${X}$, and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be ${|X|/|G|}$, since each element ${x}$ of ${X}$ will have ${|G|}$ isomorphisms on it (whether they be to the same element ${x}$, or to other elements of ${X}$).

The definition (2) can also make sense for some infinite groupoids; to my knowledge this was first explicitly done in this paper of Baez and Dolan. Consider for instance the category ${\mathbf{FinSet}}$ of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to ${\{1,\dots,n\}}$ for some natural number ${n}$, so the equivalence classes of ${\mathbf{FinSet}}$ may be indexed by the natural numbers. The automorphism group ${S_n}$ of ${\{1,\dots,n\}}$ has order ${n!}$, so the cardinality of ${\mathbf{FinSet}}$ up to isomorphism is

$\displaystyle \sum_{n=0}^\infty \frac{1}{n!} = e.$

(This fact is sometimes loosely stated as “the number of finite sets is ${e}$“, but I view this statement as somewhat misleading if the qualifier “up to isomorphism” is not added.) Similarly, when one allows for multiple isomorphisms from a group to itself, the number of groups of order four up to isomorphism is now

$\displaystyle \frac{1}{2} + \frac{1}{6} = \frac{2}{3}$

because the cyclic group ${C_4}$ has two automorphisms, whereas the Klein four-group ${C_2 \times C_2}$ has six.

In the case that the cardinality of a groupoid ${X}$ up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class ${[\mathbf{x}]}$ in ${X/\sim}$ drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class ${[x]}$ to be

$\displaystyle {\mathbf P}([\mathbf{x}] = [x]) = \frac{1 / |\mathrm{Aut}(x)|}{\sum_{[y] \in X/\sim} 1/|\mathrm{Aut}(y)|},$

thus the probability of being isomorphic to a given element ${x}$ will be inversely proportional to the number of automorphisms that ${x}$ has. For instance, if we take ${X}$ to be the set ${\{-2,-1,0,1,2\}}$ with the simply connected groupoid, ${[\mathbf{x}]}$ will be drawn uniformly from the three available equivalence classes ${[0], [1], [2]}$, with a ${1/3}$ probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of ${\{-1,+1\}}$, and draws ${[\mathbf{x}]}$ uniformly up to isomorphism, then ${[1]}$ and ${[2]}$ will now be selected with probability ${2/5}$ each, and ${[0]}$ will be selected with probability ${1/5}$. Thus this distribution has accounted for the bias mentioned previously: if a finite group ${G}$ acts on a finite space ${X}$, and ${\mathbf{x}}$ is drawn uniformly from ${X}$, then ${[\mathbf{x}]}$ now still be drawn uniformly up to isomorphism from ${X/G}$, if we use the multiply connected groupoid coming from the ${G}$ action, rather than the simply connected groupoid coming from just the ${G}$-orbit structure on ${X}$.

Using the groupoid of finite sets, we see that a finite set chosen uniformly up to isomorphism will have a cardinality that is distributed according to the Poisson distribution of parameter ${1}$, that is to say it will be of cardinality ${n}$ with probability ${\frac{e^{-1}}{n!}}$.

One important source of groupoids are the fundamental groupoids ${\pi_1(M)}$ of a manifold ${M}$ (one can also consider more general topological spaces than manifolds, but for simplicity we will restrict this discussion to the manifold case), in which the underlying space is simply ${M}$, and the isomorphisms from ${x}$ to ${y}$ are the equivalence classes of paths from ${x}$ to ${y}$ up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of ${M}$ at that base point. The equivalence class ${[x]}$ of a point in ${M}$ is then the connected component of ${x}$ in ${M}$. The cardinality up to isomorphism of the fundamental groupoid is then

$\displaystyle \sum_{M' \in \pi_0(M)} \frac{1}{|\pi_1(M')|}$

where ${\pi_0(M)}$ is the collection of connected components ${M'}$ of ${M}$, and ${|\pi_1(M')|}$ is the order of the fundamental group of ${M'}$. Thus, simply connected components of ${M}$ count for a full unit of cardinality, whereas multiply connected components (which can be viewed as quotients of their simply connected cover by their fundamental group) will count for a fractional unit of cardinality, inversely to the order of their fundamental group.

This notion of cardinality up to isomorphism of a groupoid behaves well with respect to various basic notions. For instance, suppose one has an ${n}$-fold covering map ${\pi: X \rightarrow Y}$ of one finite groupoid ${Y}$ by another ${X}$. This means that ${\pi}$ is a functor that is surjective, with all preimages of cardinality ${n}$, with the property that given any pair ${y,y'}$ in the base space ${Y}$ and any ${x}$ in the preimage ${\pi^{-1}(\{y\})}$ of ${y}$, every isomorphism ${f \in \mathrm{Iso}(y \rightarrow y')}$ has a unique lift ${\tilde f \in \mathrm{Iso}(x \rightarrow x')}$ from the given initial point ${x}$ (and some ${x'}$ in the preimage of ${y'}$). Then one can check that the cardinality up to isomorphism of ${X}$ is ${n}$ times the cardinality up to isomorphism of ${Y}$, which fits well with the geometric picture of ${X}$ as the ${n}$-fold cover of ${Y}$. (For instance, if one covers a manifold ${M}$ with finite fundamental group by its universal cover, this is a ${|\pi_1(M)|}$-fold cover, the base has cardinality ${1/|\pi_1(M)|}$ up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class ${[\mathrm{x}]}$ of ${X}$ uniformly up to isomorphism, then ${\pi([\mathrm{x}])}$ will be an equivalence class of ${Y}$ drawn uniformly up to isomorphism also.

Indeed, one can show that this notion of cardinality up to isomorphism for groupoids is uniquely determined by a small number of axioms such as these (similar to the axioms that determine Euler characteristic); see this blog post of Qiaochu Yuan for details.

The probability distributions on isomorphism classes described by the above recipe seem to arise naturally in many applications. For instance, if one draws a profinite abelian group up to isomorphism at random in this fashion (so that each isomorphism class ${[G]}$ of a profinite abelian group ${G}$ occurs with probability inversely proportional to the number of automorphisms of this group), then the resulting distribution is known as the Cohen-Lenstra distribution, and seems to emerge as the natural asymptotic distribution of many randomly generated profinite abelian groups in number theory and combinatorics, such as the class groups of random quadratic fields; see this previous blog post for more discussion. For a simple combinatorial example, the set of fixed points of a random permutation on ${n}$ elements will have a cardinality that converges in distribution to the Poisson distribution of rate ${1}$ (as discussed in this previous post), thus we see that the fixed points of a large random permutation asymptotically are distributed uniformly up to isomorphism. I’ve been told that this notion of cardinality up to isomorphism is also particularly compatible with stacks (which are a good framework to describe such objects as moduli spaces of algebraic varieties up to isomorphism), though I am not sufficiently acquainted with this theory to say much more than this.

Daniel Kane and I have just uploaded to the arXiv our paper “A bound on partitioning clusters“, submitted to the Electronic Journal of Combinatorics. In this short and elementary paper, we consider a question that arose from biomathematical applications: given a finite family ${X}$ of sets (or “clusters”), how many ways can there be of partitioning a set ${A \in X}$ in this family as the disjoint union ${A = A_1 \uplus A_2}$ of two other sets ${A_1, A_2}$ in this family? That is to say, what is the best upper bound one can place on the quantity

$\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}|$

in terms of the cardinality ${|X|}$ of ${X}$? A trivial upper bound would be ${|X|^2}$, since this is the number of possible pairs ${(A_1,A_2)}$, and ${A_1,A_2}$ clearly determine ${A}$. In our paper, we establish the improved bound

$\displaystyle | \{ (A,A_1,A_2) \in X^3: A = A_1 \uplus A_2 \}| \leq |X|^{3/p}$

where ${p}$ is the somewhat strange exponent

$\displaystyle p := \log_3 \frac{27}{4} = 1.73814\dots, \ \ \ \ \ (1)$

so that ${3/p = 1.72598\dots}$. Furthermore, this exponent is best possible!

Actually, the latter claim is quite easy to show: one takes ${X}$ to be all the subsets of ${\{1,\dots,n\}}$ of cardinality either ${n/3}$ or ${2n/3}$, for ${n}$ a multiple of ${3}$, and the claim follows readily from Stirling’s formula. So it is perhaps the former claim that is more interesting (since many combinatorial proof techniques, such as those based on inequalities such as the Cauchy-Schwarz inequality, tend to produce exponents that are rational or at least algebraic). We follow the common, though unintuitive, trick of generalising a problem to make it simpler. Firstly, one generalises the bound to the “trilinear” bound

$\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: A_3 = A_1 \uplus A_2 \}|$

$\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}$

for arbitrary finite collections ${X_1,X_2,X_3}$ of sets. One can place all the sets in ${X_1,X_2,X_3}$ inside a single finite set such as ${\{1,\dots,n\}}$, and then by replacing every set ${A_3}$ in ${X_3}$ by its complement in ${\{1,\dots,n\}}$, one can phrase the inequality in the equivalent form

$\displaystyle | \{ (A_1,A_2,A_3) \in X_1 \times X_2 \times X_3: \{1,\dots,n\} =A_1 \uplus A_2 \uplus A_3 \}|$

$\displaystyle \leq |X_1|^{1/p} |X_2|^{1/p} |X_3|^{1/p}$

for arbitrary collections ${X_1,X_2,X_3}$ of subsets of ${\{1,\dots,n\}}$. We generalise further by turning sets into functions, replacing the estimate with the slightly stronger convolution estimate

$\displaystyle f_1 * f_2 * f_3 (1,\dots,1) \leq \|f_1\|_{\ell^p(\{0,1\}^n)} \|f_2\|_{\ell^p(\{0,1\}^n)} \|f_3\|_{\ell^p(\{0,1\}^n)}$

for arbitrary functions ${f_1,f_2,f_3}$ on the Hamming cube ${\{0,1\}^n}$, where the convolution is on the integer lattice ${\bf Z}^n$ rather than on the finite field vector space ${\bf F}_2^n$. The advantage of working in this general setting is that it becomes very easy to apply induction on the dimension ${n}$; indeed, to prove this estimate for arbitrary ${n}$ it suffices to do so for ${n=1}$. This reduces matters to establishing the elementary inequality

$\displaystyle (ab(1-c))^{1/p} + (bc(1-a))^{1/p} + (ca(1-b))^{1/p} \leq 1$

for all ${0 \leq a,b,c \leq 1}$, which can be done by a combination of undergraduate multivariable calculus and a little bit of numerical computation. (The left-hand side turns out to have local maxima at ${(1,1,0), (1,0,1), (0,1,1), (2/3,2/3,2/3)}$, with the latter being the cause of the numerology (1).)

The same sort of argument also gives an energy bound

$\displaystyle E(A,A) \leq |A|^{\log_2 6}$

for any subset ${A \subset \{0,1\}^n}$ of the Hamming cube, where

$\displaystyle E(A,A) := |\{(a_1,a_2,a_3,a_4) \in A^4: a_1+a_2 = a_3 + a_4 \}|$

is the additive energy of ${A}$. The example ${A = \{0,1\}^n}$ shows that the exponent ${\log_2 6}$ cannot be improved.

I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.

One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” ${v}$ and radius ${0 < \delta < 1/2}$, define the Bohr set ${B(v,\delta)}$ to be the subset of the unit circle ${{\bf R}/{\bf Z}}$ given by the formula

$\displaystyle B(v,\delta) := \{ t \in {\bf R}/{\bf Z}: \|vt\| \leq \delta \},$

where ${\|x\|}$ denotes the distance of ${x}$ to the nearest integer. Thus, for ${v}$ positive, ${B(v,\delta)}$ is simply the union of the ${v}$ intervals ${[\frac{a-\delta}{v}, \frac{a+\delta}{v}]}$ for ${a=0,\dots,v-1}$, projected onto the unit circle ${{\bf R}/{\bf Z}}$; in the language of the usual formulation of the lonely runner conjecture, ${B(v,\delta)}$ represents those times in which a runner moving at speed ${v}$ returns to within ${\delta}$ of his or her starting position. For any non-zero integers ${v_1,\dots,v_n}$, let ${\delta(v_1,\dots,v_n)}$ be the smallest radius ${\delta}$ such that the ${n}$ Bohr sets ${B(v_1,\delta),\dots,B(v_n,\delta)}$ cover the unit circle:

$\displaystyle {\bf R}/{\bf Z} = \bigcup_{i=1}^n B(v_i,\delta). \ \ \ \ \ (1)$

Then define ${\delta_n}$ to be the smallest value of ${\delta(v_1,\dots,v_n)}$, as ${v_1,\dots,v_n}$ ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that

$\displaystyle \delta(1,\dots,n) = \frac{1}{n+1}$

and hence

$\displaystyle \delta_n \leq \frac{1}{n+1}$

for any ${n \geq 1}$. The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:

Conjecture 1 (Lonely runner conjecture) For any ${n \geq 1}$, one has ${\delta_n = \frac{1}{n+1}}$.

This conjecture is currently known for ${n \leq 6}$ (see this paper of Barajas and Serra), but remains open for higher ${n}$.

It is natural to try to attack the problem by establishing lower bounds on the quantity ${\delta_n}$. We have the following “trivial” bound, that gets within a factor of two of the conjecture:

Proposition 2 (Trivial bound) For any ${n \geq 1}$, one has ${\delta_n \geq \frac{1}{2n}}$.

Proof: It is not difficult to see that for any non-zero velocity ${v}$ and any ${0 < \delta < 1/2}$, the Bohr set ${B(v,\delta)}$ has Lebesgue measure ${m(B(v,\delta)) = 2\delta}$. In particular, by the union bound

$\displaystyle m(\bigcup_{i=1}^n B(v_i,\delta)) \leq \sum_{i=1}^n m(B(v_i,\delta)) \ \ \ \ \ (2)$

we see that the covering (1) is only possible if ${1 \leq 2 n \delta}$, giving the claim. $\Box$

So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.

Despite the crudeness of the union bound (2), it has proven surprisingly hard to make substantial improvements on the trivial bound ${\delta_n \geq \frac{1}{2n}}$. In 1994, Chen obtained the slight improvement

$\displaystyle \delta_n \geq \frac{1}{2n - 1 + \frac{1}{2n-3}}$

which was improved a little by Chen and Cusick in 1999 to

$\displaystyle \delta_n \geq \frac{1}{2n-3}$

when ${2n-3}$ was prime. In a recent paper of Perarnau and Serra, the bound

$\displaystyle \delta_n \geq \frac{1}{2n-2+o(1)}$

was obtained for arbitrary ${n}$. These bounds only improve upon the trivial bound by a multiplicative factor of ${1+O(1/n)}$. Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets ${B(v_i,\delta)}$ were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets ${B(v_i,\delta)}$ have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets ${B(v_i,\delta)}$ which are almost disjoint. For instance, suppose that we had velocities ${v_1,\dots,v_s}$ that were all prime numbers between ${n/4}$ and ${n/2}$, and that ${\delta}$ was equal to ${\delta_n}$ (and in particular was between ${1/2n}$ and ${1/(n+1)}$. Then each set ${B(v_i,\delta)}$ can be split into a “kernel” interval ${[-\frac{\delta}{v_i}, \frac{\delta}{v_i}]}$, together with the “petal” intervals ${\bigcup_{a=1}^{v_i-1} [\frac{a-\delta}{v_i}, \frac{a+\delta}{v_i}]}$. Roughly speaking, as the prime ${v_i}$ varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that

$\displaystyle m(\bigcup_{i=1}^s B(v_i,\delta)) = (1-O(\frac{1}{n})) \sum_{i=1}^s m(B(v_i,\delta)),$

so that the union bound is within a multiplicative factor of ${1+O(\frac{1}{n})}$ of the truth in this case.

This does not imply that ${\delta_n}$ is within a multiplicative factor of ${1+O(1/n)}$ of ${\frac{1}{2n}}$, though, because there are not enough primes between ${n/4}$ and ${n/2}$ to assign to ${n}$ distinct velocities; indeed, by the prime number theorem, there are only about ${\frac{n}{4\log n}}$ such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to ${\asymp n/\log n}$ Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:

Theorem 3 For sufficiently large ${n}$, one has ${\delta_n \geq \frac{1}{2n} + \frac{c \log n}{n^2 (\log\log n)^2}}$ for some absolute constant ${c>0}$.

The factors of ${\log\log n}$ in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the ${\log n}$ in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.

Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for ${\delta}$ very close to ${1/2n}$, then the multiplicity function ${\sum_{i=1}^n 1_{B(v_i,\delta)}}$ will then be mostly equal to ${1}$, but occasionally be larger than ${1}$. On the other hand, one can compute that the ${L^2}$ norm of this multiplicity function is significantly larger than ${1}$ (in fact it is at least ${(3/2-o(1))^{1/2}}$). Because of this, the ${L^3}$ norm must be very large, which means that the triple intersections ${B(v_i,\delta) \cap B(v_j,\delta) \cap B(v_k,\delta)}$ must be quite large for many triples ${(i,j,k)}$. Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities ${v_1,\dots,v_n}$ must have a large structured component, in the sense that there exists an arithmetic progression of length ${\asymp n}$ that contains ${\asymp n}$ of these velocities. For simplicity let us take the arithmetic progression to be ${\{1,\dots,n\}}$, thus ${\asymp n}$ of the velocities ${v_1,\dots,v_n}$ lie in ${\{1,\dots,n\}}$. In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between ${\log^{10} n}$ and ${n^{1/10}}$“). Using these medium-sized prime factors, one can show that many of the ${B(v_i,\delta)}$ will have quite a large overlap with many of the other ${B(v_j,\delta)}$, and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.

A modification of the above argument also allows for the improved estimate

$\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1+c-o(1)}{2n} \ \ \ \ \ (3)$

if one knows that all of the velocities ${v_1,\dots,v_n}$ are of size ${O(n)}$.

In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities ${v_1,\dots,v_n}$ are of size ${O(n^{O(n^2)})}$; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between ${O(n)}$ and ${O(n^{O(n^2)})}$, so the above bound (3) does not immediately give any new bounds for ${\delta_n}$. However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range ${O(n)}$ for which one has good results, and by decreasing the range ${O(n^{O(n^2)})}$ that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound

$\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1}{n+1} \ \ \ \ \ (4)$

holds if all the velocities ${v_1,\dots,v_n}$ are assumed to lie between ${1}$ and ${1.2 n}$. This upper threshold of ${1.2 n}$ is only a tiny improvement over the trivial threshold of ${n}$, but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to ${2n}$, as there are actually a number of ${n}$-tuples ${(v_1,\dots,v_n)}$ in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple ${(1,2,\dots,n)}$, but there is also the double ${(2,4,\dots,2n)}$ of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as ${(1,2,3,4,5,7,12)}$, or more generally one can start with the standard tuple ${(1,\dots,n)}$ and accelerate one of the velocities ${v}$ to ${2v}$; this turns out to work as long as ${v}$ shares a common factor with every integer between ${n-v+1}$ and ${2n-2v+1}$. There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length ${O(n \log n)}$ at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of ${O(n^{O(n^2)})}$ mentioned earlier was reduced all the way to something like ${O( n \log n )}$, and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.

I’ve just uploaded to the arXiv my paper “An integration approach to the Toeplitz square peg problem“, submitted to Forum of Mathematics, Sigma. This paper resulted from my attempts recently to solve the Toeplitz square peg problem (also known as the inscribed square problem):

Conjecture 1 (Toeplitz square peg problem) Let ${\gamma}$ be a simple closed curve in the plane. Is it necessarily the case that ${\gamma}$ contains four vertices of a square?

See this recent survey of Matschke in the Notices of the AMS for the latest results on this problem.

The route I took to the results in this paper was somewhat convoluted. I was motivated to look at this problem after lecturing recently on the Jordan curve theorem in my class. The problem is superficially similar to the Jordan curve theorem in that the result is known (and rather easy to prove) if ${\gamma}$ is sufficiently regular (e.g. if it is a polygonal path), but seems to be significantly more difficult when the curve is merely assumed to be continuous. Roughly speaking, all the known positive results on the problem have proceeded using (in some form or another) tools from homology: note for instance that one can view the conjecture as asking whether the four-dimensional subset ${\gamma^4}$ of the eight-dimensional space ${({\bf R}^2)^4}$ necessarily intersects the four-dimensional space ${\mathtt{Squares} \subset ({\bf R}^2)^4}$ consisting of the quadruples ${(v_1,v_2,v_3,v_4)}$ traversing a square in (say) anti-clockwise order; this space is a four-dimensional linear subspace of ${({\bf R}^2)^4}$, with a two-dimensional subspace of “degenerate” squares ${(v,v,v,v)}$ removed. If one ignores this degenerate subspace, one can use intersection theory to conclude (under reasonable “transversality” hypotheses) that ${\gamma^4}$ intersects ${\mathtt{Squares}}$ an odd number of times (up to the cyclic symmetries of the square), which is basically how Conjecture 1 is proven in the regular case. Unfortunately, if one then takes a limit and considers what happens when ${\gamma}$ is just a continuous curve, the odd number of squares created by these homological arguments could conceivably all degenerate to points, thus blocking one from proving the conjecture in the general case.

Inspired by my previous work on finite time blowup for various PDEs, I first tried looking for a counterexample in the category of (locally) self-similar curves that are smooth (or piecewise linear) away from a single origin where it can oscillate infinitely often; this is basically the smoothest type of curve that was not already covered by previous results. By a rescaling and compactness argument, it is not difficult to see that such a counterexample would exist if there was a counterexample to the following periodic version of the conjecture:

Conjecture 2 (Periodic square peg problem) Let ${\gamma_1, \gamma_2}$ be two disjoint simple closed piecewise linear curves in the cylinder ${({\bf R}/{\bf Z}) \times {\bf R}}$ which have a winding number of one, that is to say they are homologous to the loop ${x \mapsto (x,0)}$ from ${{\bf R}/{\bf Z}}$ to ${({\bf R}/{\bf Z}) \times {\bf R}}$. Then the union of ${\gamma_1}$ and ${\gamma_2}$ contains the four vertices of a square.

In contrast to Conjecture 1, which is known for polygonal paths, Conjecture 2 is still open even under the hypothesis of polygonal paths; the homological arguments alluded to previously now show that the number of inscribed squares in the periodic setting is even rather than odd, which is not enough to conclude the conjecture. (This flipping of parity from odd to even due to an infinite amount of oscillation is reminiscent of the “Eilenberg-Mazur swindle“, discussed in this previous post.)

I therefore tried to construct counterexamples to Conjecture 2. I began perturbatively, looking at curves ${\gamma_1, \gamma_2}$ that were small perturbations of constant functions. After some initial Taylor expansion, I was blocked from forming such a counterexample because an inspection of the leading Taylor coefficients required one to construct a continuous periodic function of mean zero that never vanished, which of course was impossible by the intermediate value theorem. I kept expanding to higher and higher order to try to evade this obstruction (this, incidentally, was when I discovered this cute application of Lagrange reversion) but no matter how high an accuracy I went (I think I ended up expanding to sixth order in a perturbative parameter ${\varepsilon}$ before figuring out what was going on!), this obstruction kept resurfacing again and again. I eventually figured out that this obstruction was being caused by a “conserved integral of motion” for both Conjecture 2 and Conjecture 1, which can in fact be used to largely rule out perturbative constructions. This yielded a new positive result for both conjectures:

Theorem 3

• (i) Conjecture 1 holds when ${\gamma}$ is the union ${\{ (t,f(t)): t \in [t_0,t_1]\} \cup \{ (t,g(t)): t \in [t_0,t_1]\}}$ of the graphs of two Lipschitz functions ${f,g: [t_0,t_1] \rightarrow {\bf R}}$ of Lipschitz constant less than one that agree at the endpoints.
• (ii) Conjecture 2 holds when ${\gamma_1, \gamma_2}$ are graphs of Lipschitz functions ${f: {\bf R}/{\bf Z} \rightarrow {\bf R}, g: {\bf R}/{\bf Z} \rightarrow {\bf R}}$ of Lipschitz constant less than one.

We sketch the proof of Theorem 3(i) as follows (the proof of Theorem 3(ii) is very similar). Let ${\gamma_1: [t_0, t_1] \rightarrow {\bf R}}$ be the curve ${\gamma_1(t) := (t,f(t))}$, thus ${\gamma_1}$ traverses one of the two graphs that comprise ${\gamma}$. For each time ${t \in [t_0,t_1]}$, there is a unique square with first vertex ${\gamma_1(t)}$ (and the other three vertices, traversed in anticlockwise order, denoted ${\gamma_2(t), \gamma_3(t), \gamma_4(t)}$) such that ${\gamma_2(t)}$ also lies in the graph of ${f}$ and ${\gamma_4(t)}$ also lies in the graph of ${g}$ (actually for technical reasons we have to extend ${f,g}$ by constants to all of ${{\bf R}}$ in order for this claim to be true). To see this, we simply rotate the graph of ${g}$ clockwise by ${\frac{\pi}{2}}$ around ${\gamma_1(t)}$, where (by the Lipschitz hypotheses) it must hit the graph of ${f}$ in a unique point, which is ${\gamma_2(t)}$, and which then determines the other two vertices ${\gamma_3(t), \gamma_4(t)}$ of the square. The curve ${\gamma_3(t)}$ has the same starting and ending point as the graph of ${f}$ or ${g}$; using the Lipschitz hypothesis one can show this graph is simple. If the curve ever hits the graph of ${g}$ other than at the endpoints, we have created an inscribed square, so we may assume for contradiction that ${\gamma_3(t)}$ avoids the graph of ${g}$, and hence by the Jordan curve theorem the two curves enclose some non-empty bounded open region ${\Omega}$.

Now for the conserved integral of motion. If we integrate the ${1}$-form ${y\ dx}$ on each of the four curves ${\gamma_1, \gamma_2, \gamma_3, \gamma_4}$, we obtain the identity

$\displaystyle \int_{\gamma_1} y\ dx - \int_{\gamma_2} y\ dx + \int_{\gamma_3} y\ dx - \int_{\gamma_4} y\ dx = 0.$

This identity can be established by the following calculation: one can parameterise

$\displaystyle \gamma_1(t) = (x(t), y(t))$

$\displaystyle \gamma_2(t) = (x(t)+a(t), y(t)+b(t))$

$\displaystyle \gamma_3(t) = (x(t)+a(t)-b(t), y(t)+a(t)+b(t))$

$\displaystyle \gamma_4(t) = (x(t)-b(t), y(t)+a(t))$

for some Lipschitz functions ${x,y,a,b: [t_0,t_1] \rightarrow {\bf R}}$; thus for instance ${\int_{\gamma_1} y\ dx = \int_{t_0}^{t_1} y(t)\ dx(t)}$. Inserting these parameterisations and doing some canceling, one can write the above integral as

$\displaystyle \int_{t_0}^{t_1} d \frac{a(t)^2-b(t)^2}{2}$

which vanishes because ${a(t), b(t)}$ (which represent the sidelengths of the squares determined by ${\gamma_1(t), \gamma_2(t), \gamma_3(t), \gamma_4(t)}$ vanish at the endpoints ${t=t_0,t_1}$.

Using this conserved integral of motion, one can show that

$\displaystyle \int_{\gamma_3} y\ dx = \int_{t_0}^{t_1} g(t)\ dt$

which by Stokes’ theorem then implies that the bounded open region ${\Omega}$ mentioned previously has zero area, which is absurd.

This argument hinged on the curve ${\gamma_3}$ being simple, so that the Jordan curve theorem could apply. Once one left the perturbative regime of curves of small Lipschitz constant, it became possible for ${\gamma_3}$ to be self-crossing, but nevertheless there still seemed to be some sort of integral obstruction. I eventually isolated the problem in the form of a strengthened version of Conjecture 2:

Conjecture 4 (Area formulation of square peg problem) Let ${\gamma_1, \gamma_2, \gamma_3, \gamma_4: {\bf R}/{\bf Z} \rightarrow ({\bf R}/{\bf Z}) \times {\bf R}}$ be simple closed piecewise linear curves of winding number ${1}$ obeying the area identity

$\displaystyle \int_{\gamma_1} y\ dx - \int_{\gamma_2} y\ dx + \int_{\gamma_3} y\ dx - \int_{\gamma_4} y\ dx = 0$

(note the ${1}$-form ${y\ dx}$ is still well defined on the cylinder ${({\bf R}/{\bf Z}) \times {\bf R}}$; note also that the curves ${\gamma_1,\gamma_2,\gamma_3,\gamma_4}$ are allowed to cross each other.) Then there exists a (possibly degenerate) square with vertices (traversed in anticlockwise order) lying on ${\gamma_1, \gamma_2, \gamma_3, \gamma_4}$ respectively.

It is not difficult to see that Conjecture 4 implies Conjecture 2. Actually I believe that the converse implication is at least morally true, in that any counterexample to Conjecture 4 can be eventually transformed to a counterexample to Conjecture 2 and Conjecture 1. The conserved integral of motion argument can establish Conjecture 4 in many cases, for instance if ${\gamma_2,\gamma_4}$ are graphs of functions of Lipschitz constant less than one.

Conjecture 4 has a model special case, when one of the ${\gamma_i}$ is assumed to just be a horizontal loop. In this case, the problem collapses to that of producing an intersection between two three-dimensional subsets of a six-dimensional space, rather than to four-dimensional subsets of an eight-dimensional space. More precisely, some elementary transformations reveal that this special case of Conjecture 4 can be formulated in the following fashion in which the geometric notion of a square is replaced by the additive notion of a triple of real numbers summing to zero:

Conjecture 5 (Special case of area formulation) Let ${\gamma_1, \gamma_2, \gamma_3: {\bf R}/{\bf Z} \rightarrow ({\bf R}/{\bf Z}) \times {\bf R}}$ be simple closed piecewise linear curves of winding number ${1}$ obeying the area identity

$\displaystyle \int_{\gamma_1} y\ dx + \int_{\gamma_2} y\ dx + \int_{\gamma_3} y\ dx = 0.$

Then there exist ${x \in {\bf R}/{\bf Z}}$ and ${y_1,y_2,y_3 \in {\bf R}}$ with ${y_1+y_2+y_3=0}$ such that ${(x,y_i) \in \gamma_i}$ for ${i=1,2,3}$.

This conjecture is easy to establish if one of the curves, say ${\gamma_3}$, is the graph ${\{ (t,f(t)): t \in {\bf R}/{\bf Z}\}}$ of some piecewise linear function ${f: {\bf R}/{\bf Z} \rightarrow {\bf R}}$, since in that case the curve ${\gamma_1}$ and the curve ${\tilde \gamma_2 := \{ (x, -y-f(x)): (x,y) \in \gamma_2 \}}$ enclose the same area in the sense that ${\int_{\gamma_1} y\ dx = \int_{\tilde \gamma_2} y\ dx}$, and hence must intersect by the Jordan curve theorem (otherwise they would enclose a non-zero amount of area between them), giving the claim. But when none of the ${\gamma_1,\gamma_2,\gamma_3}$ are graphs, the situation becomes combinatorially more complicated.

Using some elementary homological arguments (e.g. breaking up closed ${1}$-cycles into closed paths) and working with a generic horizontal slice of the curves, I was able to show that Conjecture 5 was equivalent to a one-dimensional problem that was largely combinatorial in nature, revolving around the sign patterns of various triple sums ${y_{1,a} + y_{2,b} + y_{3,c}}$ with ${y_{1,a}, y_{2,b}, y_{3,c}}$ drawn from various finite sets of reals.

Conjecture 6 (Combinatorial form) Let ${k_1,k_2,k_3}$ be odd natural numbers, and for each ${i=1,2,3}$, let ${y_{i,1},\dots,y_{i,k_i}}$ be distinct real numbers; we adopt the convention that ${y_{i,0}=y_{i,k_i+1}=-\infty}$. Assume the following axioms:

• (i) For any ${1 \leq p \leq k_1, 1 \leq q \leq k_2, 1 \leq r \leq k_3}$, the sums ${y_{1,p} + y_{2,q} + y_{3,r}}$ are non-zero.
• (ii) (Non-crossing) For any ${i=1,2,3}$ and ${0 \leq p < q \leq k_i}$ with the same parity, the pairs ${\{ y_{i,p}, y_{i,p+1}\}}$ and ${\{y_{i,q}, y_{i,q+1}\}}$ are non-crossing in the sense that

$\displaystyle \sum_{a \in \{p,p+1\}} \sum_{b \in \{q,q+1\}} (-1)^{a+b} \mathrm{sgn}( y_{i,a} - y_{i,b} ) = 0.$

• (iii) (Non-crossing sums) For any ${0 \leq p \leq k_1}$, ${0 \leq q \leq k_2}$, ${0 \leq r \leq k_3}$ of the same parity, one has

$\displaystyle \sum_{a \in \{p,p+1\}} \sum_{b \in \{q,q+1\}} \sum_{c \in \{r,r+1\}} (-1)^{a+b+c} \mathrm{sgn}( y_{1,a} + y_{2,b} + y_{3,c} ) = 0.$

Then one has

$\displaystyle \sum_{i=1}^3 \sum_{p=1}^{k_i} (-1)^{p-1} y_{i,p} < 0.$

Roughly speaking, Conjecture 6 and Conjecture 5 are connected by constructing curves ${\gamma_i}$ to connect ${(0, y_{i,p})}$ to ${(0,y_{i,p+1})}$ for ${0 \leq p \leq k+1}$ by various paths, which either lie to the right of the ${y}$ axis (when ${p}$ is odd) or to the left of the ${y}$ axis (when ${p}$ is even). The axiom (ii) is asserting that the numbers ${-\infty, y_{i,1},\dots,y_{i,k_i}}$ are ordered according to the permutation of a meander (formed by gluing together two non-crossing perfect matchings).

Using various ad hoc arguments involving “winding numbers”, it is possible to prove this conjecture in many cases (e.g. if one of the ${k_i}$ is at most ${3}$), to the extent that I have now become confident that this conjecture is true (and have now come full circle from trying to disprove Conjecture 1 to now believing that this conjecture holds also). But it seems that there is some non-trivial combinatorial argument to be made if one is to prove this conjecture; purely homological arguments seem to partially resolve the problem, but are not sufficient by themselves.

While I was not able to resolve the square peg problem, I think these results do provide a roadmap to attacking it, first by focusing on the combinatorial conjecture in Conjecture 6 (or its equivalent form in Conjecture 5), then after that is resolved moving on to Conjecture 4, and then finally to Conjecture 1.

[This blog post was written jointly by Terry Tao and Will Sawin.]

In the previous blog post, one of us (Terry) implicitly introduced a notion of rank for tensors which is a little different from the usual notion of tensor rank, and which (following BCCGNSU) we will call “slice rank”. This notion of rank could then be used to encode the Croot-Lev-Pach-Ellenberg-Gijswijt argument that uses the polynomial method to control capsets.

Afterwards, several papers have applied the slice rank method to further problems – to control tri-colored sum-free sets in abelian groups (BCCGNSU, KSS) and from there to the triangle removal lemma in vector spaces over finite fields (FL), to control sunflowers (NS), and to bound progression-free sets in ${p}$-groups (P).

In this post we investigate the notion of slice rank more systematically. In particular, we show how to give lower bounds for the slice rank. In many cases, we can show that the upper bounds on slice rank given in the aforementioned papers are sharp to within a subexponential factor. This still leaves open the possibility of getting a better bound for the original combinatorial problem using the slice rank of some other tensor, but for very long arithmetic progressions (at least eight terms), we show that the slice rank method cannot improve over the trivial bound using any tensor.

It will be convenient to work in a “basis independent” formalism, namely working in the category of abstract finite-dimensional vector spaces over a fixed field ${{\bf F}}$. (In the applications to the capset problem one takes ${{\bf F}={\bf F}_3}$ to be the finite field of three elements, but most of the discussion here applies to arbitrary fields.) Given ${k}$ such vector spaces ${V_1,\dots,V_k}$, we can form the tensor product ${\bigotimes_{i=1}^k V_i}$, generated by the tensor products ${v_1 \otimes \dots \otimes v_k}$ with ${v_i \in V_i}$ for ${i=1,\dots,k}$, subject to the constraint that the tensor product operation ${(v_1,\dots,v_k) \mapsto v_1 \otimes \dots \otimes v_k}$ is multilinear. For each ${1 \leq j \leq k}$, we have the smaller tensor products ${\bigotimes_{1 \leq i \leq k: i \neq j} V_i}$, as well as the ${j^{th}}$ tensor product

$\displaystyle \otimes_j: V_j \times \bigotimes_{1 \leq i \leq k: i \neq j} V_i \rightarrow \bigotimes_{i=1}^k V_i$

defined in the obvious fashion. Elements of ${\bigotimes_{i=1}^k V_i}$ of the form ${v_j \otimes_j v_{\hat j}}$ for some ${v_j \in V_j}$ and ${v_{\hat j} \in \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ will be called rank one functions, and the slice rank (or rank for short) ${\hbox{rank}(v)}$ of an element ${v}$ of ${\bigotimes_{i=1}^k V_i}$ is defined to be the least nonnegative integer ${r}$ such that ${v}$ is a linear combination of ${r}$ rank one functions. If ${V_1,\dots,V_k}$ are finite-dimensional, then the rank is always well defined as a non-negative integer (in fact it cannot exceed ${\min( \hbox{dim}(V_1), \dots, \hbox{dim}(V_k))}$. It is also clearly subadditive:

$\displaystyle \hbox{rank}(v+w) \leq \hbox{rank}(v) + \hbox{rank}(w). \ \ \ \ \ (1)$

For ${k=1}$, ${\hbox{rank}(v)}$ is ${0}$ when ${v}$ is zero, and ${1}$ otherwise. For ${k=2}$, ${\hbox{rank}(v)}$ is the usual rank of the ${2}$-tensor ${v \in V_1 \otimes V_2}$ (which can for instance be identified with a linear map from ${V_1}$ to the dual space ${V_2^*}$). The usual notion of tensor rank for higher order tensors uses complete tensor products ${v_1 \otimes \dots \otimes v_k}$, ${v_i \in V_i}$ as the rank one objects, rather than ${v_j \otimes_j v_{\hat j}}$, giving a rank that is greater than or equal to the slice rank studied here.

From basic linear algebra we have the following equivalences:

Lemma 1 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$, let ${v}$ be an element of ${V_1 \otimes \dots \otimes V_k}$, and let ${r}$ be a non-negative integer. Then the following are equivalent:

• (i) One has ${\hbox{rank}(v) \leq r}$.
• (ii) One has a representation of the form

$\displaystyle v = \sum_{j=1}^k \sum_{s \in S_j} v_{j,s} \otimes_j v_{\hat j,s}$

where ${S_1,\dots,S_k}$ are finite sets of total cardinality ${|S_1|+\dots+|S_k|}$ at most ${r}$, and for each ${1 \leq j \leq k}$ and ${s \in S_j}$, ${v_{j,s} \in V_j}$ and ${v_{\hat j,s} \in \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$.

• (iii) One has

$\displaystyle v \in \sum_{j=1}^k U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i$

where for each ${j=1,\dots,k}$, ${U_j}$ is a subspace of ${V_j}$ of total dimension ${\hbox{dim}(U_1)+\dots+\hbox{dim}(U_k)}$ at most ${r}$, and we view ${U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ as a subspace of ${\bigotimes_{i=1}^k V_i}$ in the obvious fashion.

• (iv) (Dual formulation) There exist subspaces ${W_j}$ of the dual space ${V_j^*}$ for ${j=1,\dots,k}$, of total dimension at least ${\hbox{dim}(V_1)+\dots+\hbox{dim}(V_k) - r}$, such that ${v}$ is orthogonal to ${\bigotimes_{j=1}^k W_j}$, in the sense that one has the vanishing

$\displaystyle \langle \bigotimes_{j=1}^k w_j, v \rangle = 0$

for all ${w_j \in W_j}$, where ${\langle, \rangle: \bigotimes_{j=1}^k V_j^* \times \bigotimes_{j=1}^k V_j \rightarrow {\bf F}}$ is the obvious pairing.

Proof: The equivalence of (i) and (ii) is clear from definition. To get from (ii) to (iii) one simply takes ${U_j}$ to be the span of the ${v_{j,s}}$, and conversely to get from (iii) to (ii) one takes the ${v_{j,s}}$ to be a basis of the ${U_j}$ and computes ${v_{\hat j,s}}$ by using a basis for the tensor product ${\bigotimes_{j=1}^k U_j \otimes_j \bigotimes_{1 \leq i \leq k: i \neq j} V_i}$ consisting entirely of functions of the form ${v_{j,s} \otimes_j e}$ for various ${e}$. To pass from (iii) to (iv) one takes ${W_j}$ to be the annihilator ${\{ w_j \in V_j: \langle w_j, v_j \rangle = 0 \forall v_j \in U_j \}}$ of ${U_j}$, and conversely to pass from (iv) to (iii). $\Box$

One corollary of the formulation (iv), is that the set of tensors of slice rank at most ${r}$ is Zariski closed (if the field ${{\mathbf F}}$ is algebraically closed), and so the slice rank itself is a lower semi-continuous function. This is in contrast to the usual tensor rank, which is not necessarily semicontinuous.

Corollary 2 Let ${V_1,\dots, V_k}$ be finite-dimensional vector spaces over an algebraically closed field ${{\bf F}}$. Let ${r}$ be a nonnegative integer. The set of elements of ${V_1 \otimes \dots \otimes V_k}$ of slice rank at most ${r}$ is closed in the Zariski topology.

Proof: In view of Lemma 1(i and iv), this set is the union over tuples of integers ${d_1,\dots,d_k}$ with ${d_1 + \dots + d_k \geq \hbox{dim}(V_1)+\dots+\hbox{dim}(V_k) - r}$ of the projection from ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) \times ( V_1 \otimes \dots \otimes V_k)}$ of the set of tuples ${(W_1,\dots,W_k, v)}$ with ${ v}$ orthogonal to ${W_1 \times \dots \times W_k}$, where ${\hbox{Gr}(d,V)}$ is the Grassmanian parameterizing ${d}$-dimensional subspaces of ${V}$.

One can check directly that the set of tuples ${(W_1,\dots,W_k, v)}$ with ${ v}$ orthogonal to ${W_1 \times \dots \times W_k}$ is Zariski closed in ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) \times V_1 \otimes \dots \otimes V_k}$ using a set of equations of the form ${\langle \bigotimes_{j=1}^k w_j, v \rangle = 0}$ locally on ${\hbox{Gr}(d_1, V_1) \times \dots \times \hbox{Gr}(d_k, V_k) }$. Hence because the Grassmanian is a complete variety, the projection of this set to ${V_1 \otimes \dots \otimes V_k}$ is also Zariski closed. So the finite union over tuples ${d_1,\dots,d_k}$ of these projections is also Zariski closed.

$\Box$

We also have good behaviour with respect to linear transformations:

Lemma 3 Let ${V_1,\dots,V_k, W_1,\dots,W_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$, let ${v}$ be an element of ${V_1 \otimes \dots \otimes V_k}$, and for each ${1 \leq j \leq k}$, let ${\phi_j: V_j \rightarrow W_j}$ be a linear transformation, with ${\bigotimes_{j=1}^k \phi_j: \bigotimes_{j=1}^k V_k \rightarrow \bigotimes_{j=1}^k W_k}$ the tensor product of these maps. Then

$\displaystyle \hbox{rank}( (\bigotimes_{j=1}^k \phi_j)(v) ) \leq \hbox{rank}(v). \ \ \ \ \ (2)$

Furthermore, if the ${\phi_j}$ are all injective, then one has equality in (2).

Thus, for instance, the rank of a tensor ${v \in \bigotimes_{j=1}^k V_k}$ is intrinsic in the sense that it is unaffected by any enlargements of the spaces ${V_1,\dots,V_k}$.

Proof: The bound (2) is clear from the formulation (ii) of rank in Lemma 1. For equality, apply (2) to the injective ${\phi_j}$, as well as to some arbitrarily chosen left inverses ${\phi_j^{-1}: W_j \rightarrow V_j}$ of the ${\phi_j}$. $\Box$

Computing the rank of a tensor is difficult in general; however, the problem becomes a combinatorial one if one has a suitably sparse representation of that tensor in some basis, where we will measure sparsity by the property of being an antichain.

Proposition 4 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$. For each ${1 \leq j \leq k}$, let ${(v_{j,s})_{s \in S_j}}$ be a linearly independent set in ${V_j}$ indexed by some finite set ${S_j}$. Let ${\Gamma}$ be a subset of ${S_1 \times \dots \times S_k}$.

Let ${v \in \bigotimes_{j=1}^k V_j}$ be a tensor of the form

$\displaystyle v = \sum_{(s_1,\dots,s_k) \in \Gamma} c_{s_1,\dots,s_k} v_{1,s_1} \otimes \dots \otimes v_{k,s_k} \ \ \ \ \ (3)$

where for each ${(s_1,\dots,s_k)}$, ${c_{s_1,\dots,s_k}}$ is a coefficient in ${{\bf F}}$. Then one has

$\displaystyle \hbox{rank}(v) \leq \min_{\Gamma = \Gamma_1 \cup \dots \cup \Gamma_k} |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (4)$

where the minimum ranges over all coverings of ${\Gamma}$ by sets ${\Gamma_1,\dots,\Gamma_k}$, and ${\pi_j: S_1 \times \dots \times S_k \rightarrow S_j}$ for ${j=1,\dots,k}$ are the projection maps.

Now suppose that the coefficients ${c_{s_1,\dots,s_k}}$ are all non-zero, that each of the ${S_j}$ are equipped with a total ordering ${\leq_j}$, and ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$, thus there do not exist distinct ${(s_1,\dots,s_k) \in \Gamma'}$, ${(t_1,\dots,t_k) \in \Gamma}$ such that ${s_j \leq t_j}$ for all ${j=1,\dots,k}$. Then one has

$\displaystyle \hbox{rank}(v) \geq \min_{\Gamma' = \Gamma_1 \cup \dots \cup \Gamma_k} |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)|. \ \ \ \ \ (5)$

In particular, if ${\Gamma}$ is an antichain (i.e. every element is maximal), then equality holds in (4).

Proof: By Lemma 3 (or by enlarging the bases ${v_{j,s_j}}$), we may assume without loss of generality that each of the ${V_j}$ is spanned by the ${v_{j,s_j}}$. By relabeling, we can also assume that each ${S_j}$ is of the form

$\displaystyle S_j = \{1,\dots,|S_j|\}$

with the usual ordering, and by Lemma 3 we may take each ${V_j}$ to be ${{\bf F}^{|S_j|}}$, with ${v_{j,s_j} = e_{s_j}}$ the standard basis.

Let ${r}$ denote the rank of ${v}$. To show (4), it suffices to show the inequality

$\displaystyle r \leq |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (6)$

for any covering of ${\Gamma}$ by ${\Gamma_1,\dots,\Gamma_k}$. By removing repeated elements we may assume that the ${\Gamma_i}$ are disjoint. For each ${1 \leq j \leq k}$, the tensor

$\displaystyle \sum_{(s_1,\dots,s_k) \in \Gamma_j} c_{s_1,\dots,s_k} e_{s_1} \otimes \dots \otimes e_{s_k}$

can (after collecting terms) be written as

$\displaystyle \sum_{s_j \in \pi_j(\Gamma_j)} e_{s_j} \otimes_j v_{\hat j,s_j}$

for some ${v_{\hat j, s_j} \in \bigotimes_{1 \leq i \leq k: i \neq j} {\bf F}^{|S_i|}}$. Summing and using (1), we conclude the inequality (6).

Now assume that the ${c_{s_1,\dots,s_k}}$ are all non-zero and that ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$. To conclude the proposition, it suffices to show that the reverse inequality

$\displaystyle r \geq |\pi_1(\Gamma_1)| + \dots + |\pi_k(\Gamma_k)| \ \ \ \ \ (7)$

holds for some ${\Gamma_1,\dots,\Gamma_k}$ covering ${\Gamma'}$. By Lemma 1(iv), there exist subspaces ${W_j}$ of ${({\bf F}^{|S_j|})^*}$ whose dimension ${d_j := \hbox{dim}(W_j)}$ sums to

$\displaystyle \sum_{j=1}^k d_j = \sum_{j=1}^k |S_j| - r \ \ \ \ \ (8)$

such that ${v}$ is orthogonal to ${\bigotimes_{j=1}^k W_j}$.

Let ${1 \leq j \leq k}$. Using Gaussian elimination, one can find a basis ${w_{j,1},\dots,w_{j,d_j}}$ of ${W_j}$ whose representation in the standard dual basis ${e^*_{1},\dots,e^*_{|S_j|}}$ of ${({\bf F}^{|S_j|})^*}$ is in row-echelon form. That is to say, there exist natural numbers

$\displaystyle 1 \leq s_{j,1} < \dots < s_{j,d_j} \leq |S_j|$

such that for all ${1 \leq t \leq d_j}$, ${w_{j,t}}$ is a linear combination of the dual vectors ${e^*_{s_{j,t}},\dots,e^*_{|S_j|}}$, with the ${e^*_{s_{j,t}}}$ coefficient equal to one.

We now claim that ${\prod_{j=1}^k \{ s_{j,t}: 1 \leq t \leq d_j \}}$ is disjoint from ${\Gamma'}$. Suppose for contradiction that this were not the case, thus there exists ${1 \leq t_j \leq d_j}$ for each ${1 \leq j \leq k}$ such that

$\displaystyle (s_{1,t_1}, \dots, s_{k,t_k}) \in \Gamma'.$

As ${\Gamma'}$ is the set of maximal elements of ${\Gamma}$, this implies that

$\displaystyle (s'_1,\dots,s'_k) \not \in \Gamma$

for any tuple ${(s'_1,\dots,s'_k) \in \prod_{j=1}^k \{ s_{j,t_j}, \dots, |S_j|\}}$ other than ${(s_{1,t_1}, \dots, s_{k,t_k})}$. On the other hand, we know that ${w_{j,t_j}}$ is a linear combination of ${e^*_{s_{j,t_j}},\dots,e^*_{|S_j|}}$, with the ${e^*_{s_{j,t_j}}}$ coefficient one. We conclude that the tensor product ${\bigotimes_{j=1}^k w_{j,t_j}}$ is equal to

$\displaystyle \bigotimes_{j=1}^k e^*_{s_{j,t_j}}$

plus a linear combination of other tensor products ${\bigotimes_{j=1}^k e^*_{s'_j}}$ with ${(s'_1,\dots,s'_k)}$ not in ${\Gamma}$. Taking inner products with (3), we conclude that ${\langle v, \bigotimes_{j=1}^k w_{j,t_j}\rangle = c_{s_{1,t_1},\dots,s_{k,t_k}} \neq 0}$, contradicting the fact that ${v}$ is orthogonal to ${\prod_{j=1}^k W_j}$. Thus we have ${\prod_{j=1}^k \{ s_{j,t}: 1 \leq t \leq d_j \}}$ disjoint from ${\Gamma'}$.

For each ${1 \leq j \leq k}$, let ${\Gamma_j}$ denote the set of tuples ${(s_1,\dots,s_k)}$ in ${\Gamma'}$ with ${s_j}$ not of the form ${\{ s_{j,t}: 1 \leq t \leq d_j \}}$. From the previous discussion we see that the ${\Gamma_j}$ cover ${\Gamma'}$, and we clearly have ${\pi_j(\Gamma_j) \leq |S_j| - d_j}$, and hence from (8) we have (7) as claimed. $\Box$

As an instance of this proposition, we recover the computation of diagonal rank from the previous blog post:

Example 5 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$ for some ${k \geq 2}$. Let ${d}$ be a natural number, and for ${1 \leq j \leq k}$, let ${e_{j,1},\dots,e_{j,d}}$ be a linearly independent set in ${V_j}$. Let ${c_1,\dots,c_d}$ be non-zero coefficients in ${{\bf F}}$. Then

$\displaystyle \sum_{t=1}^d c_t e_{1,t} \otimes \dots \otimes e_{k,t}$

has rank ${d}$. Indeed, one applies the proposition with ${S_1,\dots,S_k}$ all equal to ${\{1,\dots,d\}}$, with ${\Gamma}$ the diagonal in ${S_1 \times \dots \times S_k}$; this is an antichain if we give one of the ${S_i}$ the standard ordering, and another of the ${S_i}$ the opposite ordering (and ordering the remaining ${S_i}$ arbitrarily). In this case, the ${\pi_j}$ are all bijective, and so it is clear that the minimum in (4) is simply ${d}$.

The combinatorial minimisation problem in the above proposition can be solved asymptotically when working with tensor powers, using the notion of the Shannon entropy ${h(X)}$ of a discrete random variable ${X}$.

Proposition 6 Let ${V_1,\dots,V_k}$ be finite-dimensional vector spaces over a field ${{\bf F}}$. For each ${1 \leq j \leq k}$, let ${(v_{j,s})_{s \in S_j}}$ be a linearly independent set in ${V_j}$ indexed by some finite set ${S_j}$. Let ${\Gamma}$ be a non-empty subset of ${S_1 \times \dots \times S_k}$.

Let ${v \in \bigotimes_{j=1}^k V_j}$ be a tensor of the form (3) for some coefficients ${c_{s_1,\dots,s_k}}$. For each natural number ${n}$, let ${v^{\otimes n}}$ be the tensor power of ${n}$ copies of ${v}$, viewed as an element of ${\bigotimes_{j=1}^k V_j^{\otimes n}}$. Then

$\displaystyle \hbox{rank}(v^{\otimes n}) \leq \exp( (H + o(1)) n ) \ \ \ \ \ (9)$

as ${n \rightarrow \infty}$, where ${H}$ is the quantity

$\displaystyle H = \hbox{sup}_{(X_1,\dots,X_k)} \hbox{min}( h(X_1), \dots, h(X_k) ) \ \ \ \ \ (10)$

and ${(X_1,\dots,X_k)}$ range over the random variables taking values in ${\Gamma}$.

Now suppose that the coefficients ${c_{s_1,\dots,s_k}}$ are all non-zero and that each of the ${S_j}$ are equipped with a total ordering ${\leq_j}$. Let ${\Gamma'}$ be the set of maximal elements of ${\Gamma}$ in the product ordering, and let ${H' = \hbox{sup}_{(X_1,\dots,X_k)} \hbox{min}( h(X_1), \dots, h(X_k) ) }$ where ${(X_1,\dots,X_k)}$ range over random variables taking values in ${\Gamma'}$. Then

$\displaystyle \hbox{rank}(v^{\otimes n}) \geq \exp( (H' + o(1)) n ) \ \ \ \ \ (11)$

as ${n \rightarrow \infty}$. In particular, if the maximizer in (10) is supported on the maximal elements of ${\Gamma}$ (which always holds if ${\Gamma}$ is an antichain in the product ordering), then equality holds in (9).

Proof:

It will suffice to show that

$\displaystyle \min_{\Gamma^n = \Gamma_{n,1} \cup \dots \cup \Gamma_{n,k}} |\pi_{n,1}(\Gamma_{n,1})| + \dots + |\pi_{n,k}(\Gamma_{n,k})| = \exp( (H + o(1)) n ) \ \ \ \ \ (12)$

as ${n \rightarrow \infty}$, where ${\pi_{n,j}: \prod_{i=1}^k S_i^n \rightarrow S_j^n}$ is the projection map. Then the same thing will apply to ${\Gamma'}$ and ${H'}$. Then applying Proposition 4, using the lexicographical ordering on ${S_j^n}$ and noting that, if ${\Gamma'}$ are the maximal elements of ${\Gamma}$, then ${\Gamma'^n}$ are the maximal elements of ${\Gamma^n}$, we obtain both (9) and (11).

We first prove the lower bound. By compactness (and the continuity properties of entropy), we can find a random variable ${(X_1,\dots,X_k)}$ taking values in ${\Gamma}$ such that

$\displaystyle H = \hbox{min}( h(X_1), \dots, h(X_k) ). \ \ \ \ \ (13)$

Let ${\varepsilon = o(1)}$ be a small positive quantity that goes to zero sufficiently slowly with ${n}$. Let ${\Sigma = \Sigma_{X_1,\dots,X_k} \subset \Gamma^n}$ denote the set of all tuples ${(a_1, \dots, \vec a_n)}$ in ${\Gamma^n}$ that are within ${\varepsilon}$ of being distributed according to the law of ${(X_1,\dots,X_k)}$, in the sense that for all ${a \in \Gamma}$, one has

$\displaystyle |\frac{|\{ 1 \leq l \leq n: a_l = a \}|}{n} - {\bf P}( (X_1,\dots,X_k) = a )| \leq \varepsilon.$

By the asymptotic equipartition property, the cardinality of ${\Sigma}$ can be computed to be

$\displaystyle |\Sigma| = \exp( (h( X_1,\dots,X_k)+o(1)) n ) \ \ \ \ \ (14)$

if ${\varepsilon}$ goes to zero slowly enough. Similarly one has

$\displaystyle |\pi_{n,j}(\Sigma)| = \exp( (h( X_j)+o(1)) n ),$

and for each ${s_{n,j} \in \pi_{n,j}(\Sigma)}$, one has

$\displaystyle |\{ \sigma \in \Sigma: \pi_{n,j}(\sigma) = s_{n,j} \}| \leq \exp( (h( X_1,\dots,X_k)-h(X_j)+o(1)) n ). \ \ \ \ \ (15)$

Now let ${\Gamma^n = \Gamma_{n,1} \cup \dots \cup \Gamma_{n,k}}$ be an arbitrary covering of ${\Gamma^n}$. By the pigeonhole principle, there exists ${1 \leq j \leq k}$ such that

$\displaystyle |\Gamma_{n,j} \cap \Sigma| \geq \frac{1}{k} |\Sigma|$

and hence by (14), (15)

$\displaystyle |\pi_{n,j}( \Gamma_{n,j} \cap \Sigma)| \geq \frac{1}{k} \exp( (h( X_j)+o(1)) n )$

which by (13) implies that

$\displaystyle |\pi_{n,1}(\Gamma_{n,1})| + \dots + |\pi_{n,k}(\Gamma_{n,k})| \geq \exp( (H + o(1)) n )$

noting that the ${\frac{1}{k}}$ factor can be absorbed into the ${o(1)}$ error). This gives the lower bound in (12).

Now we prove the upper bound. We can cover ${\Gamma^n}$ by ${O(\exp(o(n))}$ sets of the form ${\Sigma_{X_1,\dots,X_k}}$ for various choices of random variables ${(X_1,\dots,X_k)}$ taking values in ${\Gamma}$. For each such random variable ${(X_1,\dots,X_k)}$, we can find ${1 \leq j \leq k}$ such that ${h(X_j) \leq H}$; we then place all of ${\Sigma_{X_1,\dots,X_k}}$ in ${\Gamma_j}$. It is then clear that the ${\Gamma_j}$ cover ${\Gamma}$ and that

$\displaystyle |\Gamma_j| \leq \exp( (H+o(1)) n )$

for all ${j=1,\dots,n}$, giving the required upper bound. $\Box$

It is of interest to compute the quantity ${H}$ in (10). We have the following criterion for when a maximiser occurs:

Proposition 7 Let ${S_1,\dots,S_k}$ be finite sets, and ${\Gamma \subset S_1 \times \dots \times S_k}$ be non-empty. Let ${H}$ be the quantity in (10). Let ${(X_1,\dots,X_k)}$ be a random variable taking values in ${\Gamma}$, and let ${\Gamma^* \subset \Gamma}$ denote the essential range of ${(X_1,\dots,X_k)}$, that is to say the set of tuples ${(t_1,\dots,t_k)\in \Gamma}$ such that ${{\bf P}( X_1=t_1, \dots, X_k = t_k)}$ is non-zero. Then the following are equivalent:

• (i) ${(X_1,\dots,X_k)}$ attains the maximum in (10).
• (ii) There exist weights ${w_1,\dots,w_k \geq 0}$ and a finite quantity ${D \geq 0}$, such that ${w_j=0}$ whenever ${h(X_j) > \min(h(X_1),\dots,h(X_k))}$, and such that

$\displaystyle \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)} \leq D \ \ \ \ \ (16)$

for all ${(t_1,\dots,t_k) \in \Gamma}$, with equality if ${(t_1,\dots,t_k) \in \Gamma^*}$. (In particular, ${w_j}$ must vanish if there exists a ${t_j \in \pi_i(\Gamma)}$ with ${{\bf P}(X_j=t_j)=0}$.)

Furthermore, when (i) and (ii) holds, one has

$\displaystyle D = H \sum_{j=1}^k w_j. \ \ \ \ \ (17)$

Proof: We first show that (i) implies (ii). The function ${p \mapsto p \log \frac{1}{p}}$ is concave on ${[0,1]}$. As a consequence, if we define ${C}$ to be the set of tuples ${(h_1,\dots,h_k) \in [0,+\infty)^k}$ such that there exists a random variable ${(Y_1,\dots,Y_k)}$ taking values in ${\Gamma}$ with ${h(Y_j) \geq h_j}$, then ${C}$ is convex. On the other hand, by (10), ${C}$ is disjoint from the orthant ${(H,+\infty)^k}$. Thus, by the hyperplane separation theorem, we conclude that there exists a half-space

$\displaystyle \{ (h_1,\dots,h_k) \in {\bf R}^k: w_1 h_1 + \dots + w_k h_k \geq c \},$

where ${w_1,\dots,w_k}$ are reals that are not all zero, and ${c}$ is another real, which contains ${(h(X_1),\dots,h(X_k))}$ on its boundary and ${(H,+\infty)^k}$ in its interior, such that ${C}$ avoids the interior of the half-space. Since ${(h(X_1),\dots,h(X_k))}$ is also on the boundary of ${(H,+\infty)^k}$, we see that the ${w_j}$ are non-negative, and that ${w_j = 0}$ whenever ${h(X_j) \neq H}$.

By construction, the quantity

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k)$

is maximised when ${(Y_1,\dots,Y_k) = (X_1,\dots,X_k)}$. At this point we could use the method of Lagrange multipliers to obtain the required constraints, but because we have some boundary conditions on the ${(Y_1,\dots,Y_k)}$ (namely, that the probability that they attain a given element of ${\Gamma}$ has to be non-negative) we will work things out by hand. Let ${t = (t_1,\dots,t_k)}$ be an element of ${\Gamma}$, and ${s = (s_1,\dots,s_k)}$ an element of ${\Gamma^*}$. For ${\varepsilon>0}$ small enough, we can form a random variable ${(Y_1,\dots,Y_k)}$ taking values in ${\Gamma}$, whose probability distribution is the same as that for ${(X_1,\dots,X_k)}$ except that the probability of attaining ${(t_1,\dots,t_k)}$ is increased by ${\varepsilon}$, and the probability of attaining ${(s_1,\dots,s_k)}$ is decreased by ${\varepsilon}$. If there is any ${j}$ for which ${{\bf P}(X_j = t_j)=0}$ and ${w_j \neq 0}$, then one can check that

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k) - (w_1 h(X_1) + \dots + w_k h(X_k)) \gg \varepsilon \log \frac{1}{\varepsilon}$

for sufficiently small ${\varepsilon}$, contradicting the maximality of ${(X_1,\dots,X_k)}$; thus we have ${{\bf P}(X_j = t_j) > 0}$ whenever ${w_j \neq 0}$. Taylor expansion then gives

$\displaystyle w_1 h(Y_1) + \dots + w_k h(Y_k) - (w_1 h(X_1) + \dots + w_k h(X_k)) = (A_t - A_s) \varepsilon + O(\varepsilon^2)$

for small ${\varepsilon}$, where

$\displaystyle A_t := \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)}$

and similarly for ${A_s}$. We conclude that ${A_t \leq A_s}$ for all ${s \in \Gamma^*}$ and ${t \in \Gamma}$, thus there exists a quantity ${D}$ such that ${A_s = D}$ for all ${s \in \Gamma^*}$, and ${A_t \leq D}$ for all ${t \in \Gamma}$. By construction ${D}$ must be nonnegative. Sampling ${(t_1,\dots,t_k)}$ using the distribution of ${(X_1,\dots,X_k)}$, one has

$\displaystyle \sum_{j=1}^k w_j \log \frac{1}{{\bf P}(X_j = t_j)} = D$

almost surely; taking expectations we conclude that

$\displaystyle \sum_{j=1}^k w_j \sum_{t_j \in S_j} {\bf P}( X_j = t_j) \log \frac{1}{{\bf P}(X_j = t_j)} = D.$

The inner sum is ${h(X_j)}$, which equals ${H}$ when ${w_j}$ is non-zero, giving (17).

Now we show conversely that (ii) implies (i). As noted previously, the function ${p \mapsto p \log \frac{1}{p}}$ is concave on ${[0,1]}$, with derivative ${\log \frac{1}{p} - 1}$. This gives the inequality

$\displaystyle q \log \frac{1}{q} \leq p \log \frac{1}{p} + (q-p) ( \log \frac{1}{p} - 1 ) \ \ \ \ \ (18)$

for any ${0 \leq p,q \leq 1}$ (note the right-hand side may be infinite when ${p=0}$ and ${q>0}$). Let ${(Y_1,\dots,Y_k)}$ be any random variable taking values in ${\Gamma}$, then on applying the above inequality with ${p = {\bf P}(X_j = t_j)}$ and ${q = {\bf P}( Y_j = t_j )}$, multiplying by ${w_j}$, and summing over ${j=1,\dots,k}$ and ${t_j \in S_j}$ gives

$\displaystyle \sum_{j=1}^k w_j h(Y_j) \leq \sum_{j=1}^k w_j h(X_j)$

$\displaystyle + \sum_{j=1}^k \sum_{t_j \in S_j} w_j ({\bf P}(Y_j = t_j) - {\bf P}(X_j = t_j)) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 ).$

By construction, one has

$\displaystyle \sum_{j=1}^k w_j h(X_j) = \min(h(X_1),\dots,h(X_k)) \sum_{j=1}^k w_j$

and

$\displaystyle \sum_{j=1}^k w_j h(Y_j) \geq \min(h(Y_1),\dots,h(Y_k)) \sum_{j=1}^k w_j$

so to prove that ${\min(h(Y_1),\dots,h(Y_k)) \leq \min(h(X_1),\dots,h(X_k))}$ (which would give (i)), it suffices to show that

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j ({\bf P}(Y_j = t_j) - {\bf P}(X_j = t_j)) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 ) \leq 0,$

or equivalently that the quantity

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) ( \log \frac{1}{{\bf P}(X_j=t_j)} - 1 )$

is maximised when ${(Y_1,\dots,Y_k) = (X_1,\dots,X_k)}$. Since

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) = \sum_{j=1}^k w_j$

it suffices to show this claim for the quantity

$\displaystyle \sum_{j=1}^k \sum_{t_j \in S_j} w_j {\bf P}(Y_j = t_j) \log \frac{1}{{\bf P}(X_j=t_j)}.$

One can view this quantity as

$\displaystyle {\bf E}_{(Y_1,\dots,Y_k)} \sum_{j=1}^k w_j \log \frac{1}{{\bf P}_{X_j}(X_j=Y_j)}.$

By (ii), this quantity is bounded by ${D}$, with equality if ${(Y_1,\dots,Y_k)}$ is equal to ${(X_1,\dots,X_k)}$ (and is in particular ranging in ${\Gamma^*}$), giving the claim. $\Box$

The second half of the proof of Proposition 7 only uses the marginal distributions ${{{\bf P}(X_j=t_j)}}$ and the equation(16), not the actual distribution of ${(X_1,\dots,X_k)}$, so it can also be used to prove an upper bound on ${H}$ when the exact maximizing distribution is not known, given suitable probability distributions in each variable. The logarithm of the probability distribution here plays the role that the weight functions do in BCCGNSU.

Remark 8 Suppose one is in the situation of (i) and (ii) above; assume the nondegeneracy condition that ${H}$ is positive (or equivalently that ${D}$ is positive). We can assign a “degree” ${d_j(t_j)}$ to each element ${t_j \in S_j}$ by the formula

$\displaystyle d_j(t_j) := w_j \log \frac{1}{{\bf P}(X_j = t_j)}, \ \ \ \ \ (19)$

then every tuple ${(t_1,\dots,t_k)}$ in ${\Gamma}$ has total degree at most ${D}$, and those tuples in ${\Gamma^*}$ have degree exactly ${D}$. In particular, every tuple in ${\Gamma^n}$ has degree at most ${nD}$, and hence by (17), each such tuple has a ${j}$-component of degree less than or equal to ${nHw_j}$ for some ${j}$ with ${w_j>0}$. On the other hand, we can compute from (19) and the fact that ${h(X_j) = H}$ for ${w_j > 0}$ that ${Hw_j = {\bf E} d_j(X_j)}$. Thus, by asymptotic equipartition, and assuming ${w_j \neq 0}$, the number of “monomials” in ${S_j^n}$ of total degree at most ${nHw_j}$ is at most ${\exp( (h(X_j)+o(1)) n )}$; one can in fact use (19) and (18) to show that this is in fact an equality. This gives a direct way to cover ${\Gamma^n}$ by sets ${\Gamma_{n,1},\dots,\Gamma_{n,k}}$ with ${|\pi_j(\Gamma_{n,j})| \leq \exp( (H+o(1)) n)}$, which is in the spirit of the Croot-Lev-Pach-Ellenberg-Gijswijt arguments from the previous post.

We can now show that the rank computation for the capset problem is sharp:

Proposition 9 Let ${V_1^{\otimes n} = V_2^{\otimes n} = V_3^{\otimes n}}$ denote the space of functions from ${{\bf F}_3^n}$ to ${{\bf F}_3}$. Then the function ${(x,y,z) \mapsto \delta_{0^n}(x,y,z)}$ from ${{\bf F}_3^n \times {\bf F}_3^n \times {\bf F}_3^n}$ to ${{\bf F}}$, viewed as an element of ${V_1^{\otimes n} \otimes V_2^{\otimes n} \otimes V_3^{\otimes n}}$, has rank ${\exp( (H^*+o(1)) n )}$ as ${n \rightarrow \infty}$, where ${H^* \approx 1.013445}$ is given by the formula

$\displaystyle H^* = \alpha \log \frac{1}{\alpha} + \beta \log \frac{1}{\beta} + \gamma \log \frac{1}{\gamma} \ \ \ \ \ (20)$

with

$\displaystyle \alpha = \frac{32}{3(15 + \sqrt{33})} \approx 0.51419$

$\displaystyle \beta = \frac{4(\sqrt{33}-1)}{3(15+\sqrt{33})} \approx 0.30495$

$\displaystyle \gamma = \frac{(\sqrt{33}-1)^2}{6(15+\sqrt{33})} \approx 0.18086.$

Proof: In ${{\bf F}_3 \times {\bf F}_3 \times {\bf F}_3}$, we have

$\displaystyle \delta_0(x+y+z) = 1 - (x+y+z)^2$

$\displaystyle = (1-x^2) - y^2 - z^2 + xy + yz + zx.$

Thus, if we let ${V_1=V_2=V_3}$ be the space of functions from ${{\bf F}_3}$ to ${{\bf F}_3}$ (with domain variable denoted ${x,y,z}$ respectively), and define the basis functions

$\displaystyle v_{1,0} := 1; v_{1,1} := x; v_{1,2} := x^2$

$\displaystyle v_{2,0} := 1; v_{2,1} := y; v_{2,2} := y^2$

$\displaystyle v_{3,0} := 1; v_{3,1} := z; v_{3,2} := z^2$

of ${V_1,V_2,V_3}$ indexed by ${S_1=S_2=S_3 := \{ 0,1,2\}}$ (with the usual ordering), respectively, and set ${\Gamma \subset S_1 \times S_2 \times S_3}$ to be the set

$\displaystyle \{ (2,0,0), (0,2,0), (0,0,2), (1,1,0), (0,1,1), (1,0,1),(0,0,0) \}$

then ${\delta_0(x,y,z)}$ is a linear combination of the ${v_{1,t_1} \otimes v_{1,t_2} \otimes v_{1,t_3}}$ with ${(t_1,t_2,t_3) \in \Gamma}$, and all coefficients non-zero. Then we have ${\Gamma'= \{ (2,0,0), (0,2,0), (0,0,2), (1,1,0), (0,1,1), (1,0,1) \}}$. We will show that the quantity ${H}$ of (10) agrees with the quantity ${H^*}$ of (20), and that the optimizing distribution is supported on ${\Gamma'}$, so that by Proposition 6 the rank of ${\delta_{0^n}(x,y,z)}$ is ${\exp( (H+o(1)) n)}$.

To compute the quantity at (10), we use the criterion in Proposition 7. We take ${(X_1,X_2,X_3)}$ to be the random variable taking values in ${\Gamma}$ that attains each of the values ${(2,0,0), (0,2,0), (0,0,2)}$ with a probability of ${\gamma \approx 0.18086}$, and each of ${(1,1,0), (0,1,1), (1,0,1)}$ with a probability of ${\alpha - 2\gamma = \beta/2 \approx 0.15247}$; then each of the ${X_j}$ attains the values of ${0,1,2}$ with probabilities ${\alpha,\beta,\gamma}$ respectively, so in particular ${h(X_1)=h(X_2)=h(X_3)}$ is equal to the quantity ${H'}$ in (20). If we now set ${w_1 = w_2 = w_3 := 1}$ and

$\displaystyle D := 2\log \frac{1}{\alpha} + \log \frac{1}{\gamma} = \log \frac{1}{\alpha} + 2 \log \frac{1}{\beta} = 3H^* \approx 3.04036$

we can verify the condition (16) with equality for all ${(t_1,t_2,t_3) \in \Gamma'}$, which from (17) gives ${H=H'=H^*}$ as desired. $\Box$

This statement already follows from the result of Kleinberg-Sawin-Speyer, which gives a “tri-colored sum-free set” in ${\mathbb F_3^n}$ of size ${\exp((H'+o(1))n)}$, as the slice rank of this tensor is an upper bound for the size of a tri-colored sum-free set. If one were to go over the proofs more carefully to evaluate the subexponential factors, this argument would give a stronger lower bound than KSS, as it does not deal with the substantial loss that comes from Behrend’s construction. However, because it actually constructs a set, the KSS result rules out more possible approaches to give an exponential improvement of the upper bound for capsets. The lower bound on slice rank shows that the bound cannot be improved using only the slice rank of this particular tensor, whereas KSS shows that the bound cannot be improved using any method that does not take advantage of the “single-colored” nature of the problem.

We can also show that the slice rank upper bound in a result of Naslund-Sawin is similarly sharp:

Proposition 10 Let ${V_1^{\otimes n} = V_2^{\otimes n} = V_3^{\otimes n}}$ denote the space of functions from ${\{0,1\}^n}$ to ${\mathbb C}$. Then the function ${(x,y,z) \mapsto \prod_{i=1}^n (x_i+y_i+z_i)-1}$ from ${\{0,1\}^n \times \{0,1\}^n \times \{0,1\}^n \rightarrow \mathbb C}$, viewed as an element of ${V_1^{\otimes n} \otimes V_2^{\otimes n} \otimes V_3^{\otimes n}}$, has slice rank ${(3/2^{2/3})^n e^{o(n)}}$

Proof: Let ${v_{1,0}=1}$ and ${v_{1,1}=x}$ be a basis for the space ${V_1}$ of functions on ${\{0,1\}}$, itself indexed by ${S_1=\{0,1\}}$. Choose similar bases for ${V_2}$ and ${V_3}$, with ${v_{2,0}=1, v_{2,1}=y}$ and ${v_{3,0}=1,v_{3,1}=z-1}$.

Set ${\Gamma = \{(1,0,0),(0,1,0),(0,0,1)\}}$. Then ${x+y+z-1}$ is a linear combination of the ${v_{1,t_1} \otimes v_{1,t_2} \otimes v_{1,t_3}}$ with ${(t_1,t_2,t_3) \in \Gamma}$, and all coefficients non-zero. Order ${S_1,S_2,S_3}$ the usual way so that ${\Gamma}$ is an antichain. We will show that the quantity ${H}$ of (10) is ${\log(3/2^{2/3})}$, so that applying the last statement of Proposition 6, we conclude that the rank of ${\delta_{0^n}(x,y,z)}$ is ${\exp( (\log(3/2^{2/3})+o(1)) n)= (3/2^{2/3})^n e^{o(n)}}$ ,

Let ${(X_1,X_2,X_3)}$ be the random variable taking values in ${\Gamma}$ that attains each of the values ${(1,0,0),(0,1,0),(0,0,1)}$ with a probability of ${1/3}$. Then each of the ${X_i}$ attains the value ${1}$ with probability ${1/3}$ and ${0}$ with probability ${2/3}$, so

$\displaystyle h(X_1)=h(X_2)=h(X_3) = (1/3) \log (3) + (2/3) \log(3/2) = \log 3 - (2/3) \log 2= \log (3/2^{2/3})$

Setting ${w_1=w_2=w_3=1}$ and ${D=3 \log(3/2^{2/3})=3 \log 3 - 2 \log 2}$, we can verify the condition (16) with equality for all ${(t_1,t_2,t_3) \in \Gamma'}$, which from (17) gives ${H=\log (3/2^{2/3})}$ as desired. $\Box$

We used a slightly different method in each of the last two results. In the first one, we use the most natural bases for all three vector spaces, and distinguish ${\Gamma}$ from its set of maximal elements ${\Gamma'}$. In the second one we modify one basis element slightly, with ${v_{3,1}=z-1}$ instead of the more obvious choice ${z}$, which allows us to work with ${\Gamma = \{(1,0,0),(0,1,0),(0,0,1)\}}$ instead of ${\Gamma=\{(1,0,0),(0,1,0),(0,0,1),(0,0,0)\}}$. Because ${\Gamma}$ is an antichain, we do not need to distinguish ${\Gamma}$ and ${\Gamma'}$. Both methods in fact work with either problem, and they are both about equally difficult, but we include both as either might turn out to be substantially more convenient in future work.

Proposition 11 Let ${k \geq 8}$ be a natural number and let ${G}$ be a finite abelian group. Let ${{\bf F}}$ be any field. Let ${V_1 = \dots = V_k}$ denote the space of functions from ${G}$ to ${{\bf F}}$.

Let ${F}$ be any ${{\bf F}}$-valued function on ${G^k}$ that is nonzero only when the ${k}$ elements of ${G^n}$ form a ${k}$-term arithmetic progression, and is nonzero on every ${k}$-term constant progression.

Then the slice rank of ${F}$ is ${|G|}$.

Proof: We apply Proposition 4, using the standard bases of ${V_1,\dots,V_k}$. Let ${\Gamma}$ be the support of ${F}$. Suppose that we have ${k}$ orderings on ${H}$ such that the constant progressions are maximal elements of ${\Gamma}$ and thus all constant progressions lie in ${\Gamma'}$. Then for any partition ${\Gamma_1,\dots, \Gamma_k}$ of ${\Gamma'}$, ${\Gamma_j}$ can contain at most ${|\pi_j(\Gamma_j)|}$ constant progressions, and as all ${|G|}$ constant progressions must lie in one of the ${\Gamma_j}$, we must have ${\sum_{j=1}^k |\pi_j(\Gamma_j)| \geq |G|}$. By Proposition 4, this implies that the slice rank of ${F}$ is at least ${|G|}$. Since ${F}$ is a ${|G| \times \dots \times |G|}$ tensor, the slice rank is at most ${|G|}$, hence exactly ${|G|}$.

So it is sufficient to find ${k}$ orderings on ${G}$ such that the constant progressions are maximal element of ${\Gamma}$. We make several simplifying reductions: We may as well assume that ${\Gamma}$ consists of all the ${k}$-term arithmetic progressions, because if the constant progressions are maximal among the set of all progressions then they are maximal among its subset ${\Gamma}$. So we are looking for an ordering in which the constant progressions are maximal among all ${k}$-term arithmetic progressions. We may as well assume that ${G}$ is cyclic, because if for each cyclic group we have an ordering where constant progressions are maximal, on an arbitrary finite abelian group the lexicographic product of these orderings is an ordering for which the constant progressions are maximal. We may assume ${k=8}$, as if we have an ${8}$-tuple of orderings where constant progressions are maximal, we may add arbitrary orderings and the constant progressions will remain maximal.

So it is sufficient to find ${8}$ orderings on the cyclic group ${\mathbb Z/n}$ such that the constant progressions are maximal elements of the set of ${8}$-term progressions in ${\mathbb Z/n}$ in the ${8}$-fold product ordering. To do that, let the first, second, third, and fifth orderings be the usual order on ${\{0,\dots,n-1\}}$ and let the fourth, sixth, seventh, and eighth orderings be the reverse of the usual order on ${\{0,\dots,n-1\}}$.

Then let ${(c,c,c,c,c,c,c,c)}$ be a constant progression and for contradiction assume that ${(a,a+b,a+2b,a+3b,a+4b,a+5b,a+6b,a+7b)}$ is a progression greater than ${(c,c,c,c,c,c,c,c)}$ in this ordering. We may assume that ${c \in [0, (n-1)/2]}$, because otherwise we may reverse the order of the progression, which has the effect of reversing all eight orderings, and then apply the transformation ${x \rightarrow n-1-x}$, which again reverses the eight orderings, bringing us back to the original problem but with ${c \in [0,(n-1)/2]}$.

Take a representative of the residue class ${b}$ in the interval ${[-n/2,n/2]}$. We will abuse notation and call this ${b}$. Observe that ${a+b, a+2b,}$ ${a+3b}$, and ${a+5b}$ are all contained in the interval ${[0,c]}$ modulo ${n}$. Take a representative of the residue class ${a}$ in the interval ${[0,c]}$. Then ${a+b}$ is in the interval ${[mn,mn+c]}$ for some ${m}$. The distance between any distinct pair of intervals of this type is greater than ${n/2}$, but the distance between ${a}$ and ${a+b}$ is at most ${n/2}$, so ${a+b}$ is in the interval ${[0,c]}$. By the same reasoning, ${a+2b}$ is in the interval ${[0,c]}$. Therefore ${|b| \leq c/2< n/4}$. But then the distance between ${a+2b}$ and ${a+4b}$ is at most ${n/2}$, so by the same reasoning ${a+4b}$ is in the interval ${[0,c]}$. Because ${a+3b}$ is between ${a+2b}$ and ${a+4b}$, it also lies in the interval ${[0,c]}$. Because ${a+3b}$ is in the interval ${[0,c]}$, and by assumption it is congruent mod ${n}$ to a number in the set ${\{0,\dots,n-1\}}$ greater than or equal to ${c}$, it must be exactly ${c}$. Then, remembering that ${a+2b}$ and ${a+4b}$ lie in ${[0,c]}$, we have ${c-b \leq b}$ and ${c+b \leq b}$, so ${b=0}$, hence ${a=c}$, thus ${(a,\dots,a+7b)=(c,\dots,c)}$, which contradicts the assumption that ${(a,\dots,a+7b)>(c,\dots,c)}$. $\Box$

In fact, given a ${k}$-term progressions mod ${n}$ and a constant, we can form a ${k}$-term binary sequence with a ${1}$ for each step of the progression that is greater than the constant and a ${0}$ for each step that is less. Because a rotation map, viewed as a dynamical system, has zero topological entropy, the number of ${k}$-term binary sequences that appear grows subexponentially in ${k}$. Hence there must be, for large enough ${k}$, at least one sequence that does not appear. In this proof we exploit a sequence that does not appear for ${k=8}$.