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A capset in the vector space {{\bf F}_3^n} over the finite field {{\bf F}_3} of three elements is a subset {A} of {{\bf F}_3^n} that does not contain any lines {\{ x,x+r,x+2r\}}, where {x,r \in {\bf F}_3^n} and {r \neq 0}. A basic problem in additive combinatorics (discussed in one of the very first posts on this blog) is to obtain good upper and lower bounds for the maximal size of a capset in {{\bf F}_3^n}.

Trivially, one has {|A| \leq 3^n}. Using Fourier methods (and the density increment argument of Roth), the bound of {|A| \leq O( 3^n / n )} was obtained by Meshulam, and improved only as late as 2012 to {O( 3^n /n^{1+c})} for some absolute constant {c>0} by Bateman and Katz. But in a very recent breakthrough, Ellenberg (and independently Gijswijt) obtained the exponentially superior bound {|A| \leq O( 2.756^n )}, using a version of the polynomial method recently introduced by Croot, Lev, and Pach. (In the converse direction, a construction of Edel gives capsets as large as {(2.2174)^n}.) Given the success of the polynomial method in superficially similar problems such as the finite field Kakeya problem (discussed in this previous post), it was natural to wonder that this method could be applicable to the cap set problem (see for instance this MathOverflow comment of mine on this from 2010), but it took a surprisingly long time before Croot, Lev, and Pach were able to identify the precise variant of the polynomial method that would actually work here.

The proof of the capset bound is very short (Ellenberg’s and Gijswijt’s preprints are both 3 pages long, and Croot-Lev-Pach is 6 pages), but I thought I would present a slight reformulation of the argument which treats the three points on a line in {{\bf F}_3} symmetrically (as opposed to treating the third point differently from the first two, as is done in the Ellenberg and Gijswijt papers; Croot-Lev-Pach also treat the middle point of a three-term arithmetic progression differently from the two endpoints, although this is a very natural thing to do in their context of {({\bf Z}/4{\bf Z})^n}). The basic starting point is this: if {A} is a capset, then one has the identity

\displaystyle \delta_{0^n}( x+y+z ) = \sum_{a \in A} \delta_a(x) \delta_a(y) \delta_a(z) \ \ \ \ \ (1)

 

for all {(x,y,z) \in A^3}, where {\delta_a(x) := 1_{a=x}} is the Kronecker delta function, which we view as taking values in {{\bf F}_3}. Indeed, (1) reflects the fact that the equation {x+y+z=0} has solutions precisely when {x,y,z} are either all equal, or form a line, and the latter is ruled out precisely when {A} is a capset.

To exploit (1), we will show that the left-hand side of (1) is “low rank” in some sense, while the right-hand side is “high rank”. Recall that a function {F: A \times A \rightarrow {\bf F}} taking values in a field {{\bf F}} is of rank one if it is non-zero and of the form {(x,y) \mapsto f(x) g(y)} for some {f,g: A \rightarrow {\bf F}}, and that the rank of a general function {F: A \times A \rightarrow {\bf F}} is the least number of rank one functions needed to express {F} as a linear combination. More generally, if {k \geq 2}, we define the rank of a function {F: A^k \rightarrow {\bf F}} to be the least number of “rank one” functions of the form

\displaystyle (x_1,\dots,x_k) \mapsto f(x_i) g(x_1,\dots,x_{i-1},x_{i+1},\dots,x_k)

for some {i=1,\dots,k} and some functions {f: A \rightarrow {\bf F}}, {g: A^{k-1} \rightarrow {\bf F}}, that are needed to generate {F} as a linear combination. For instance, when {k=3}, the rank one functions take the form {(x,y,z) \mapsto f(x) g(y,z)}, {(x,y,z) \mapsto f(y) g(x,z)}, {(x,y,z) \mapsto f(z) g(x,y)}, and linear combinations of {r} such rank one functions will give a function of rank at most {r}.

It is a standard fact in linear algebra that the rank of a diagonal matrix is equal to the number of non-zero entries. This phenomenon extends to higher dimensions:

Lemma 1 (Rank of diagonal hypermatrices) Let {k \geq 2}, let {A} be a finite set, let {{\bf F}} be a field, and for each {a \in A}, let {c_a \in {\bf F}} be a coefficient. Then the rank of the function

\displaystyle (x_1,\dots,x_k) \mapsto \sum_{a \in A} c_a \delta_a(x_1) \dots \delta_a(x_k) \ \ \ \ \ (2)

 

is equal to the number of non-zero coefficients {c_a}.

Proof: We induct on {k}. As mentioned above, the case {k=2} follows from standard linear algebra, so suppose now that {k>2} and the claim has already been proven for {k-1}.

It is clear that the function (2) has rank at most equal to the number of non-zero {c_a} (since the summands on the right-hand side are rank one functions), so it suffices to establish the lower bound. By deleting from {A} those elements {a \in A} with {c_a=0} (which cannot increase the rank), we may assume without loss of generality that all the {c_a} are non-zero. Now suppose for contradiction that (2) has rank at most {|A|-1}, then we obtain a representation

\displaystyle \sum_{a \in A} c_a \delta_a(x_1) \dots \delta_a(x_k)

\displaystyle = \sum_{i=1}^k \sum_{\alpha \in I_i} f_{i,\alpha}(x_i) g_{i,\alpha}( x_1,\dots,x_{i-1},x_{i+1},\dots,x_k) \ \ \ \ \ (3)

 

for some sets {I_1,\dots,I_k} of cardinalities adding up to at most {|A|-1}, and some functions {f_{i,\alpha}: A \rightarrow {\bf F}} and {g_{i,\alpha}: A^{k-1} \rightarrow {\bf R}}.

Consider the space of functions {h: A \rightarrow {\bf F}} that are orthogonal to all the {f_{k,\alpha}}, {\alpha \in I_k} in the sense that

\displaystyle \sum_{x \in A} f_{k,\alpha}(x) h(x) = 0

for all {\alpha \in I_k}. This space is a vector space whose dimension {d} is at least {|A| - |I_k|}. A basis of this space generates a {d \times |A|} coordinate matrix of full rank, which implies that there is at least one non-singular {d \times d} minor. This implies that there exists a function {h: A \rightarrow {\bf F}} in this space which is nowhere vanishing on some subset {A'} of {A} of cardinality at least {|A|-|I_k|}.

If we multiply (3) by {h(x_k)} and sum in {x_k}, we conclude that

\displaystyle \sum_{a \in A} c_a h(a) \delta_a(x_1) \dots \delta_a(x_{k-1})

\displaystyle = \sum_{i=1}^{k-1} \sum_{\alpha \in I_i} f_{i,\alpha}(x_i)\tilde g_{i,\alpha}( x_1,\dots,x_{i-1},x_{i+1},\dots,x_{k-1})

where

\displaystyle \tilde g_{i,\alpha}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_{k-1})

\displaystyle := \sum_{x_k \in A} g_{i,\alpha}(x_1,\dots,x_{i-1},x_{i+1},\dots,x_k) h(x_k).

The right-hand side has rank at most {|A|-1-|I_k|}, since the summands are rank one functions. On the other hand, from induction hypothesis the left-hand side has rank at least {|A|-|I_k|}, giving the required contradiction. \Box

On the other hand, we have the following (symmetrised version of a) beautifully simple observation of Croot, Lev, and Pach:

Lemma 2 On {({\bf F}_3^n)^3}, the rank of the function {(x,y,z) \mapsto \delta_{0^n}(x+y+z)} is at most {3N}, where

\displaystyle N := \sum_{a,b,c \geq 0: a+b+c=n, b+2c \leq 2n/3} \frac{n!}{a!b!c!}.

Proof: Using the identity {\delta_0(x) = 1 - x^2} for {x \in {\bf F}_3}, we have

\displaystyle \delta_{0^n}(x+y+z) = \prod_{i=1}^n (1 - (x_i+y_i+z_i)^2).

The right-hand side is clearly a polynomial of degree {2n} in {x,y,z}, which is then a linear combination of monomials

\displaystyle x_1^{i_1} \dots x_n^{i_n} y_1^{j_1} \dots y_n^{j_n} z_1^{k_1} \dots z_n^{k_n}

with {i_1,\dots,i_n,j_1,\dots,j_n,k_1,\dots,k_n \in \{0,1,2\}} with

\displaystyle i_1 + \dots + i_n + j_1 + \dots + j_n + k_1 + \dots + k_n \leq 2n.

In particular, from the pigeonhole principle, at least one of {i_1 + \dots + i_n, j_1 + \dots + j_n, k_1 + \dots + k_n} is at most {2n/3}.

Consider the contribution of the monomials for which {i_1 + \dots + i_n \leq 2n/3}. We can regroup this contribution as

\displaystyle \sum_\alpha f_\alpha(x) g_\alpha(y,z)

where {\alpha} ranges over those {(i_1,\dots,i_n) \in \{0,1,2\}^n} with {i_1 + \dots + i_n \leq 2n/3}, {f_\alpha} is the monomial

\displaystyle f_\alpha(x_1,\dots,x_n) := x_1^{i_1} \dots x_n^{i_n}

and {g_\alpha: {\bf F}_3^n \times {\bf F}_3^n \rightarrow {\bf F}_3} is some explicitly computable function whose exact form will not be of relevance to our argument. The number of such {\alpha} is equal to {N}, so this contribution has rank at most {N}. The remaining contributions arising from the cases {j_1 + \dots + j_n \leq 2n/3} and {k_1 + \dots + k_n \leq 2n/3} similarly have rank at most {N} (grouping the monomials so that each monomial is only counted once), so the claim follows.

Upon restricting from {({\bf F}_3^n)^3} to {A^3}, the rank of {(x,y,z) \mapsto \delta_{0^n}(x+y+z)} is still at most {3N}. The two lemmas then combine to give the Ellenberg-Gijswijt bound

\displaystyle |A| \leq 3N.

All that remains is to compute the asymptotic behaviour of {N}. This can be done using the general tool of Cramer’s theorem, but can also be derived from Stirling’s formula (discussed in this previous post). Indeed, if {a = (\alpha+o(1)) n}, {b = (\beta+o(1)) n}, {c = (\gamma+o(1)) n} for some {\alpha,\beta,\gamma \geq 0} summing to {1}, Stirling’s formula gives

\displaystyle \frac{n!}{a!b!c!} = \exp( n (h(\alpha,\beta,\gamma) + o(1)) )

where {h} is the entropy function

\displaystyle h(\alpha,\beta,\gamma) = \alpha \log \frac{1}{\alpha} + \beta \log \frac{1}{\beta} + \gamma \log \frac{1}{\gamma}.

We then have

\displaystyle N = \exp( n (X + o(1))

where {X} is the maximum entropy {h(\alpha,\beta,\gamma)} subject to the constraints

\displaystyle \alpha,\beta,\gamma \geq 0; \alpha+\beta+\gamma=1; \beta+2\gamma \leq 2/3.

A routine Lagrange multiplier computation shows that the maximum occurs when

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

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

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

and {h(\alpha,\beta,\gamma)} is approximately {1.013455}, giving rise to the claimed bound of {O( 2.756^n )}.

Remark 3 As noted in the Ellenberg and Gijswijt papers, the above argument extends readily to other fields than {{\bf F}_3} to control the maximal size of subset of {{\bf F}^n} that has no non-trivial solutions to the equation {ax+by+cz=0}, where {a,b,c \in {\bf F}} are non-zero constants that sum to zero. Of course one replaces the function {(x,y,z) \mapsto \delta_{0^n}(x+y+z)} in Lemma 2 by {(x,y,z) \mapsto \delta_{0^n}(ax+by+cz)} in this case.

Remark 4 This symmetrised formulation suggests that one possible way to improve slightly on the numerical quantity {2.756} by finding a more efficient way to decompose {\delta_{0^n}(x+y+z)} into rank one functions, however I was not able to do so (though such improvements are reminiscent of the Strassen type algorithms for fast matrix multiplication).

Remark 5 It is tempting to see if this method can get non-trivial upper bounds for sets {A} with no length {4} progressions, in (say) {{\bf F}_5^n}. One can run the above arguments, replacing the function

\displaystyle (x,y,z) \mapsto \delta_{0^n}(x+y+z)

with

\displaystyle (x,y,z,w) \mapsto \delta_{0^n}(x-2y+z) \delta_{0^n}(y-2z+w);

this leads to the bound {|A| \leq 4N} where

\displaystyle N := \sum_{a,b,c,d,e \geq 0: a+b+c+d+e=n, b+2c+3d+4e \leq 2n} \frac{n!}{a!b!c!d!e!}.

Unfortunately, {N} is asymptotic to {\frac{1}{2} 5^n} and so this bound is in fact slightly worse than the trivial bound {|A| \leq 5^n}! However, there is a slim chance that there is a more efficient way to decompose {\delta_{0^n}(x-2y+z) \delta_{0^n}(y-2z+w)} into rank one functions that would give a non-trivial bound on {A}. I experimented with a few possible such decompositions but unfortunately without success.

Remark 6 Return now to the capset problem. Since Lemma 1 is valid for any field {{\bf F}}, one could perhaps hope to get better bounds by viewing the Kronecker delta function {\delta} as taking values in another field than {{\bf F}_3}, such as the complex numbers {{\bf C}}. However, as soon as one works in a field of characteristic other than {3}, one can adjoin a cube root {\omega} of unity, and one now has the Fourier decomposition

\displaystyle \delta_{0^n}(x+y+z) = \frac{1}{3^n} \sum_{\xi \in {\bf F}_3^n} \omega^{\xi \cdot x} \omega^{\xi \cdot y} \omega^{\xi \cdot z}.

Moving to the Fourier basis, we conclude from Lemma 1 that the function {(x,y,z) \mapsto \delta_{0^n}(x+y+z)} on {{\bf F}_3^n} now has rank exactly {3^n}, and so one cannot improve upon the trivial bound of {|A| \leq 3^n} by this method using fields of characteristic other than three as the range field. So it seems one has to stick with {{\bf F}_3} (or the algebraic completion thereof).

Thanks to Jordan Ellenberg and Ben Green for helpful discussions.

I’ve just uploaded to the arXiv my paper “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures“, submitted to the Festschrift “Number Theory – Diophantine problems, uniform distribution and applications” in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) conjectures in the two-point case. In that paper, the estimate

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+h)}{n} = o( \log x )

as {x \rightarrow \infty} was demonstrated, where {h} was any positive integer and {\lambda} denoted the Liouville function. The proof proceeded using a method I call the “entropy decrement argument”, which ultimately reduced matters to establishing a bound of the form

\displaystyle  \sum_{n \leq x} \frac{|\sum_{h \leq H} \lambda(n+h) e( \alpha h)|}{n} = o( H \log x )

whenever {H} was a slowly growing function of {x}. This was in turn established in a previous paper of Matomaki, Radziwill, and myself, using the recent breakthrough of Matomaki and Radziwill.

It is natural to see to what extent the arguments can be adapted to attack the higher-point cases of the logarithmically averaged Chowla conjecture (ignoring for this post the more general Elliott conjecture for other bounded multiplicative functions than the Liouville function). That is to say, one would like to prove that

\displaystyle  \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o( \log x )

as {x \rightarrow \infty} for any fixed distinct integers {h_1,\dots,h_k}. As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form

\displaystyle  \sum_{n \leq x} \frac{1}{n} \| \lambda \|_{U^d[n, n+H]} = o( \log x )

for {H} a slowly growing function of {x} and some fixed {d} (in fact we can take {d=k-1} for {k \geq 3}), where {U^d} is the (normalised) local Gowers uniformity norm. (In the case {k=3}, {d=2}, this becomes the Fourier-uniformity conjecture discussed in this previous post.) If one then applied the (now proven) inverse conjecture for the Gowers norms, this estimate is in turn equivalent to the more complicated looking assertion

\displaystyle  \sum_{n \leq x} \frac{1}{n} \sup |\sum_{h \leq H} \lambda(n+h) F( g^h x )| = o( \log x ) \ \ \ \ \ (1)

where the supremum is over all possible choices of nilsequences {h \mapsto F(g^h x)} of controlled step and complexity (see the paper for definitions of these terms).

The main novelty in the paper (elaborating upon a previous comment I had made on this blog) is to observe that this latter estimate in turn follows from the logarithmically averaged form of Sarnak’s conjecture (discussed in this previous post), namely that

\displaystyle  \sum_{n \leq x} \frac{1}{n} \lambda(n) F( T^n x )= o( \log x )

whenever {n \mapsto F(T^n x)} is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in (1) means that we need a little bit more; roughly speaking, we need the class of nilsequences of a given step and complexity to have “uniformly zero entropy” in some sense.

On the other hand, it was already known (see previous post) that the Chowla conjecture implied the Sarnak conjecture, and similarly for the logarithmically averaged form of the two conjectures. Putting all these implications together, we obtain the pleasant fact that the logarithmically averaged Sarnak and Chowla conjectures are equivalent, which is the main result of the current paper. There have been a large number of special cases of the Sarnak conjecture worked out (when the deterministic sequence involved came from a special dynamical system), so these results can now also be viewed as partial progress towards the Chowla conjecture also (at least with logarithmic averaging). However, my feeling is that the full resolution of these conjectures will not come from these sorts of special cases; instead, conjectures like the Fourier-uniformity conjecture in this previous post look more promising to attack.

It would also be nice to get rid of the pesky logarithmic averaging, but this seems to be an inherent requirement of the entropy decrement argument method, so one would probably have to find a way to avoid that argument if one were to remove the log averaging.

When teaching mathematics, the traditional method of lecturing in front of a blackboard is still hard to improve upon, despite all the advances in modern technology.  However, there are some nice things one can do in an electronic medium, such as this blog.  Here, I would like to experiment with the ability to animate images, which I think can convey some mathematical concepts in ways that cannot be easily replicated by traditional static text and images. Given that many readers may find these animations annoying, I am placing the rest of the post below the fold.

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Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold {M} can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space {{\bf R}^d}. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let {M} be a compact manifold. Then there exists an embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In fact, if {M} is {n}-dimensional, one can take {d} to equal {2n}, which is often best possible (easy examples include the circle {{\bf R}/{\bf Z}} which embeds into {{\bf R}^2} but not {{\bf R}^1}, or the Klein bottle that embeds into {{\bf R}^4} but not {{\bf R}^3}). One can also relax the compactness hypothesis on {M} to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take {d} equal to {2n+1}.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let {(M,g)} be a compact Riemannian manifold. Then there exists a isometric embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In order to obtain the isometric embedding, the dimension {d} has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

\displaystyle  d = \max( 	n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)

is attained, which I believe is still the record for large {n}. (In the converse direction, one cannot do better than {d = \frac{n(n+1)}{2}}, basically because this is the number of degrees of freedom in the Riemannian metric {g}.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of {d}, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in {d} is that it allows one to freely exploit the very useful tool of pairing together two maps {u_1: M \rightarrow {\bf R}^{d_1}}, {u_2: M \rightarrow {\bf R}^{d_2}} to form a combined map {(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}} that can be closer to an embedding or an isometric embedding than the original maps {u_1,u_2}. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of {M} and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

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Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms.  This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function \zeta(s) and the discriminant modular form \Delta(q), but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.

As of today, the “L-functions and modular forms database” is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops.  An article by John Cremona summarising the purpose of the database can be found here.

(Thanks to Andrew Sutherland and Kiran Kedlaya for the information.)

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