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Van Vu and I just posted to the arXiv our paper “sum-free sets in groups” (submitted to Discrete Analysis), as well as a companion survey article (submitted to J. Comb.). Given a subset {A} of an additive group {G = (G,+)}, define the quantity {\phi(A)} to be the cardinality of the largest subset {B} of {A} which is sum-free in {A} in the sense that all the sums {b_1+b_2} with {b_1,b_2} distinct elements of {B} lie outside of {A}. For instance, if {A} is itself a group, then {\phi(A)=1}, since no two elements of {A} can sum to something outside of {A}. More generally, if {A} is the union of {k} groups, then {\phi(A)} is at most {k}, thanks to the pigeonhole principle.

If {G} is the integers, then there are no non-trivial subgroups, and one can thus expect {\phi(A)} to start growing with {A}. For instance, one has the following easy result:

Proposition 1 Let {A} be a set of {2^k} natural numbers. Then {\phi(A) > k}.

Proof: We use an argument of Ruzsa, which is based in turn on an older argument of Choi. Let {x_1} be the largest element of {A}, and then recursively, once {x_1,\dots,x_i} has been selected, let {x_{i+1}} be the largest element of {A} not equal to any of the {x_1,\dots,x_i}, such that {x_{i+1}+x_j \not \in A} for all {j=1,\dots,i}, terminating this construction when no such {x_{i+1}} can be located. This gives a sequence {x_1 > x_2 > \dots > x_m} of elements in {A} which are sum-free in {A}, and with the property that for any {y \in A}, either {y} is equal to one of the {x_i}, or else {y + x_i \in A} for some {i} with {x_i > y}. Iterating this, we see that any {y \in A} is of the form {x_{i_1} - x_{i_2} - \dots - x_{i_j}} for some {j \geq 1} and {1 \leq i_1 < i_2 < \dots \leq i_j \leq m}. The number of such expressions {x_{i_1} - x_{i_2} - \dots - x_{i_j}} is at most {2^{m}-1}, thus {2^k \leq 2^m-1} which implies {m \geq k+1}. Since {\phi(A) \geq m}, the claim follows. \Box

In particular, we have {\phi(A) \gg \log |A|} for subsets {A} of the integers. It has been possible to improve upon this easy bound, but only with remarkable effort. The best lower bound currently is

\displaystyle \phi(A) \geq \log |A| (\log\log|A|)^{1/2 - o(1)},

a result of Shao (building upon earlier work of Sudakov, Szemeredi, and Vu and of Dousse). In the opposite direction, a construction of Ruzsa gives examples of large sets {A} with {\phi(A) \leq \exp( O( \sqrt{\log |A|} ) )}.

Using the standard tool of Freiman homomorphisms, the above results for the integers extend to other torsion-free abelian groups {G}. In our paper we study the opposite case where {G} is finite (but still abelian). In this paper of Erdös (in which the quantity {\phi(A)} was first introduced), the following question was posed: if {A} is sufficiently large depending on {\phi(A)}, does this imply the existence of two elements {x,y \in A} with {x+y=0}? As it turns out, we were able to find some simple counterexamples to this statement. For instance, if {H} is any finite additive group, then the set {A := \{ 1 \hbox{ mod } 7, 2 \hbox{ mod } 7, 4 \hbox{ mod } 7\} \times H \subset {\bf Z}/7{\bf Z} \times H} has {\phi(A)=3} but with no {x,y \in A} summing to zero; this type of example in fact works with {7} replaced by any larger Mersenne prime, and we also have a counterexample in {{\bf Z}/2^n{\bf Z}} for {n} arbitrarily large. However, in the positive direction, we can show that the answer to Erdös’s question is positive if {|G|} is assumed to have no small prime factors. That is to say,

Theorem 2 For every {k \geq 1} there exists {C \geq 1} such that if {G} is a finite abelian group whose order is not divisible by any prime less than or equal to {C}, and {A} is a subset of {G} with order at least {C} and {\phi(A) \leq k}, then there exist {x,y \in A} with {x+y=0}.

There are two main tools used to prove this result. One is an “arithmetic removal lemma” proven by Král, Serra, and Vena. Note that the condition {\phi(A) \leq k} means that for any distinct {x_1,\dots,x_{k+1} \in A}, at least one of the {x_i+x_j}, {1 \leq i < j \leq k+1}, must also lie in {A}. Roughly speaking, the arithmetic removal lemma allows one to “almost” remove the requirement that {x_1,\dots,x_{k+1}} be distinct, which basically now means that {x \in A \implies 2x \in A} for almost all {x \in A}. This near-dilation symmetry, when combined with the hypothesis that {|G|} has no small prime factors, gives a lot of “dispersion” in the Fourier coefficients of {1_A} which can now be exploited to prove the theorem.

The second tool is the following structure theorem, which is the main result of our paper, and goes a fair ways towards classifying sets {A} for which {\phi(A)} is small:

Theorem 3 Let {A} be a finite subset of an arbitrary additive group {G}, with {\phi(A) \leq k}. Then one can find finite subgroups {H_1,\dots,H_m} with {m \leq k} such that {|A \cap H_i| \gg_k |H_i|} and {|A \backslash (H_1 \cup \dots \cup H_m)| \ll_k 1}. Furthermore, if {m=k}, then the exceptional set {A \backslash (H_1 \cup \dots \cup H_m)} is empty.

Roughly speaking, this theorem shows that the example of the union of {k} subgroups mentioned earlier is more or less the “only” example of sets {A} with {\phi(A) \leq k}, modulo the addition of some small exceptional sets and some refinement of the subgroups to dense subsets.

This theorem has the flavour of other inverse theorems in additive combinatorics, such as Freiman’s theorem, and indeed one can use Freiman’s theorem (and related tools, such as the Balog-Szemeredi theorem) to easily get a weaker version of this theorem. Indeed, if there are no sum-free subsets of {A} of order {k+1}, then a fraction {\gg_k 1} of all pairs {a,b} in {A} must have their sum also in {A} (otherwise one could take {k+1} random elements of {A} and they would be sum-free in {A} with positive probability). From this and the Balog-Szemeredi theorem and Freiman’s theorem (in arbitrary abelian groups, as established by Green and Ruzsa), we see that {A} must be “commensurate” with a “coset progression” {H+P} of bounded rank. One can then eliminate the torsion-free component {P} of this coset progression by a number of methods (e.g. by using variants of the argument in Proposition 1), with the upshot being that one can locate a finite group {H_1} that has large intersection with {A}.

At this point it is tempting to simply remove {H_1} from {A} and iterate. But one runs into a technical difficulty that removing a set such as {H_1} from {A} can alter the quantity {\phi(A)} in unpredictable ways, so one has to still keep {H_1} around when analysing the residual set {A \backslash H_1}. A second difficulty is that the latter set {A \backslash H_1} could be considerably smaller than {A} or {H_1}, but still large in absolute terms, so in particular any error term whose size is only bounded by {\varepsilon |A|} for a small {\varepsilon} could be massive compared with the residual set {A\backslash H_1}, and so such error terms would be unacceptable. One can get around these difficulties if one first performs some preliminary “normalisation” of the group {H_1}, so that the residual set {A \backslash H_1} does not intersect any coset of {H_1} too strongly. The arguments become even more complicated when one starts removing more than one group {H_1,\dots,H_i} from {A} and analyses the residual set {A \backslash (H_1 \cup \dots \cup H_i)}; indeed the “epsilon management” involved became so fearsomely intricate that we were forced to use a nonstandard analysis formulation of the problem in order to keep the complexity of the argument at a reasonable level (cf. my previous blog post on this topic). One drawback of doing so is that we have no effective bounds for the implied constants in our main theorem; it would be of interest to obtain a more direct proof of our main theorem that would lead to effective bounds.

I’ve just uploaded two related papers to the arXiv:

This pair of papers is an outgrowth of these two recent blog posts and the ensuing discussion. In the first paper, we establish the following logarithmically averaged version of the Chowla conjecture (in the case {k=2} of two-point correlations (or “pair correlations”)):

Theorem 1 (Logarithmically averaged Chowla conjecture) Let {a_1,a_2} be natural numbers, and let {b_1,b_2} be integers such that {a_1 b_2 - a_2 b_1 \neq 0}. Let {1 \leq \omega(x) \leq x} be a quantity depending on {x} that goes to infinity as {x \rightarrow \infty}. Let {\lambda} denote the Liouville function. Then one has

\displaystyle  \sum_{x/\omega(x) < n \leq x} \frac{\lambda(a_1 n + b_1) \lambda(a_2 n+b_2)}{n} = o( \log \omega(x) ) \ \ \ \ \ (1)

as {x \rightarrow \infty}.

Thus for instance one has

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n} = o(\log x). \ \ \ \ \ (2)

For comparison, the non-averaged Chowla conjecture would imply that

\displaystyle  \sum_{n \leq x} \lambda(n) \lambda(n+1) = o(x) \ \ \ \ \ (3)

which is a strictly stronger estimate than (2), and remains open.

The arguments also extend to other completely multiplicative functions than the Liouville function. In particular, one obtains a slightly averaged version of the non-asymptotic Elliott conjecture that was shown in the previous blog post to imply a positive solution to the Erdos discrepancy problem. The averaged version of the conjecture established in this paper is slightly weaker than the one assumed in the previous blog post, but it turns out that the arguments there can be modified without much difficulty to accept this averaged Elliott conjecture as input. In particular, we obtain an unconditional solution to the Erdos discrepancy problem as a consequence; this is detailed in the second paper listed above. In fact we can also handle the vector-valued version of the Erdos discrepancy problem, in which the sequence {f(1), f(2), \dots} takes values in the unit sphere of an arbitrary Hilbert space, rather than in {\{-1,+1\}}.

Estimates such as (2) or (3) are known to be subject to the “parity problem” (discussed numerous times previously on this blog), which roughly speaking means that they cannot be proven solely using “linear” estimates on functions such as the von Mangoldt function. However, it is known that the parity problem can be circumvented using “bilinear” estimates, and this is basically what is done here.

We now describe in informal terms the proof of Theorem 1, focusing on the model case (2) for simplicity. Suppose for contradiction that the left-hand side of (2) was large and (say) positive. Using the multiplicativity {\lambda(pn) = -\lambda(n)}, we conclude that

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+p) 1_{p|n}}{n}

is also large and positive for all primes {p} that are not too large; note here how the logarithmic averaging allows us to leave the constraint {n \leq x} unchanged. Summing in {p}, we conclude that

\displaystyle  \sum_{n \leq x} \frac{ \sum_{p \in {\mathcal P}} \lambda(n) \lambda(n+p) 1_{p|n}}{n}

is large and positive for any given set {{\mathcal P}} of medium-sized primes. By a standard averaging argument, this implies that

\displaystyle  \frac{1}{H} \sum_{j=1}^H \sum_{p \in {\mathcal P}} \lambda(n+j) \lambda(n+p+j) 1_{p|n+j} \ \ \ \ \ (4)

is large for many choices of {n}, where {H} is a medium-sized parameter at our disposal to choose, and we take {{\mathcal P}} to be some set of primes that are somewhat smaller than {H}. (A similar approach was taken in this recent paper of Matomaki, Radziwill, and myself to study sign patterns of the Möbius function.) To obtain the required contradiction, one thus wants to demonstrate significant cancellation in the expression (4). As in that paper, we view {n} as a random variable, in which case (4) is essentially a bilinear sum of the random sequence {(\lambda(n+1),\dots,\lambda(n+H))} along a random graph {G_{n,H}} on {\{1,\dots,H\}}, in which two vertices {j, j+p} are connected if they differ by a prime {p} in {{\mathcal P}} that divides {n+j}. A key difficulty in controlling this sum is that for randomly chosen {n}, the sequence {(\lambda(n+1),\dots,\lambda(n+H))} and the graph {G_{n,H}} need not be independent. To get around this obstacle we introduce a new argument which we call the “entropy decrement argument” (in analogy with the “density increment argument” and “energy increment argument” that appear in the literature surrounding Szemerédi’s theorem on arithmetic progressions, and also reminiscent of the “entropy compression argument” of Moser and Tardos, discussed in this previous post). This argument, which is a simple consequence of the Shannon entropy inequalities, can be viewed as a quantitative version of the standard subadditivity argument that establishes the existence of Kolmogorov-Sinai entropy in topological dynamical systems; it allows one to select a scale parameter {H} (in some suitable range {[H_-,H_+]}) for which the sequence {(\lambda(n+1),\dots,\lambda(n+H))} and the graph {G_{n,H}} exhibit some weak independence properties (or more precisely, the mutual information between the two random variables is small).

Informally, the entropy decrement argument goes like this: if the sequence {(\lambda(n+1),\dots,\lambda(n+H))} has significant mutual information with {G_{n,H}}, then the entropy of the sequence {(\lambda(n+1),\dots,\lambda(n+H'))} for {H' > H} will grow a little slower than linearly, due to the fact that the graph {G_{n,H}} has zero entropy (knowledge of {G_{n,H}} more or less completely determines the shifts {G_{n+kH,H}} of the graph); this can be formalised using the classical Shannon inequalities for entropy (and specifically, the non-negativity of conditional mutual information). But the entropy cannot drop below zero, so by increasing {H} as necessary, at some point one must reach a metastable region (cf. the finite convergence principle discussed in this previous blog post), within which very little mutual information can be shared between the sequence {(\lambda(n+1),\dots,\lambda(n+H))} and the graph {G_{n,H}}. Curiously, for the application it is not enough to have a purely quantitative version of this argument; one needs a quantitative bound (which gains a factor of a bit more than {\log H} on the trivial bound for mutual information), and this is surprisingly delicate (it ultimately comes down to the fact that the series {\sum_{j \geq 2} \frac{1}{j \log j \log\log j}} diverges, which is only barely true).

Once one locates a scale {H} with the low mutual information property, one can use standard concentration of measure results such as the Hoeffding inequality to approximate (4) by the significantly simpler expression

\displaystyle  \frac{1}{H} \sum_{j=1}^H \sum_{p \in {\mathcal P}} \frac{\lambda(n+j) \lambda(n+p+j)}{p}. \ \ \ \ \ (5)

The important thing here is that Hoeffding’s inequality gives exponentially strong bounds on the failure probability, which is needed to counteract the logarithms that are inevitably present whenever trying to use entropy inequalities. The expression (5) can then be controlled in turn by an application of the Hardy-Littlewood circle method and a non-trivial estimate

\displaystyle  \sup_\alpha \frac{1}{X} \int_X^{2X} |\frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(1) \ \ \ \ \ (6)

for averaged short sums of a modulated Liouville function established in another recent paper by Matomäki, Radziwill and myself.

When one uses this method to study more general sums such as

\displaystyle  \sum_{n \leq x} \frac{g_1(n) g_2(n+1)}{n},

one ends up having to consider expressions such as

\displaystyle  \frac{1}{H} \sum_{j=1}^H \sum_{p \in {\mathcal P}} c_p \frac{g_1(n+j) g_2(n+p+j)}{p}.

where {c_p} is the coefficient {c_p := \overline{g_1}(p) \overline{g_2}(p)}. When attacking this sum with the circle method, one soon finds oneself in the situation of wanting to locate the large Fourier coefficients of the exponential sum

\displaystyle  S(\alpha) := \sum_{p \in {\mathcal P}} \frac{c_p}{p} e^{2\pi i \alpha p}.

In many cases (such as in the application to the Erdös discrepancy problem), the coefficient {c_p} is identically {1}, and one can understand this sum satisfactorily using the classical results of Vinogradov: basically, {S(\alpha)} is large when {\alpha} lies in a “major arc” and is small when it lies in a “minor arc”. For more general functions {g_1,g_2}, the coefficients {c_p} are more or less arbitrary; the large values of {S(\alpha)} are no longer confined to the major arc case. Fortunately, even in this general situation one can use a restriction theorem for the primes established some time ago by Ben Green and myself to show that there are still only a bounded number of possible locations {\alpha} (up to the uncertainty mandated by the Heisenberg uncertainty principle) where {S(\alpha)} is large, and we can still conclude by using (6). (Actually, as recently pointed out to me by Ben, one does not need the full strength of our result; one only needs the {L^4} restriction theorem for the primes, which can be proven fairly directly using Plancherel’s theorem and some sieve theory.)

It is tempting to also use the method to attack higher order cases of the (logarithmically) averaged Chowla conjecture, for instance one could try to prove the estimate

\displaystyle  \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1) \lambda(n+2)}{n} = o(\log x).

The above arguments reduce matters to obtaining some non-trivial cancellation for sums of the form

\displaystyle  \frac{1}{H} \sum_{j=1}^H \sum_{p \in {\mathcal P}} \frac{\lambda(n+j) \lambda(n+p+j) \lambda(n+2p+j)}{p}.

A little bit of “higher order Fourier analysis” (as was done for very similar sums in the ergodic theory context by Frantzikinakis-Host-Kra and Wooley-Ziegler) lets one control this sort of sum if one can establish a bound of the form

\displaystyle  \frac{1}{X} \int_X^{2X} \sup_\alpha |\frac{1}{H} \sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(1) \ \ \ \ \ (7)

where {X} goes to infinity and {H} is a very slowly growing function of {X}. This looks very similar to (6), but the fact that the supremum is now inside the integral makes the problem much more difficult. However it looks worth attacking (7) further, as this estimate looks like it should have many nice applications (beyond just the {k=3} case of the logarithmically averaged Chowla or Elliott conjectures, which is already interesting).

For higher {k} than {k=3}, the same line of analysis requires one to replace the linear phase {e(\alpha n)} by more complicated phases, such as quadratic phases {e(\alpha n^2 + \beta n)} or even {k-2}-step nilsequences. Given that (7) is already beyond the reach of current literature, these even more complicated expressions are also unavailable at present, but one can imagine that they will eventually become tractable, in which case we would obtain an averaged form of the Chowla conjecture for all {k}, which would have a number of consequences (such as a logarithmically averaged version of Sarnak’s conjecture, as per this blog post).

It would of course be very nice to remove the logarithmic averaging, and be able to establish bounds such as (3). I did attempt to do so, but I do not see a way to use the entropy decrement argument in a manner that does not require some sort of averaging of logarithmic type, as it requires one to pick a scale {H} that one cannot specify in advance, which is not a problem for logarithmic averages (which are quite stable with respect to dilations) but is problematic for ordinary averages. But perhaps the problem can be circumvented by some clever modification of the argument. One possible approach would be to start exploiting multiplicativity at products of primes, and not just individual primes, to try to keep the scale fixed, but this makes the concentration of measure part of the argument much more complicated as one loses some independence properties (coming from the Chinese remainder theorem) which allowed one to conclude just from the Hoeffding inequality.

This week I have been at a Banff workshop “Combinatorics meets Ergodic theory“, focused on the combinatorics surrounding Szemerédi’s theorem and the Gowers uniformity norms on one hand, and the ergodic theory surrounding Furstenberg’s multiple recurrence theorem and the Host-Kra structure theory on the other. This was quite a fruitful workshop, and directly inspired the various posts this week on this blog. Incidentally, BIRS being as efficient as it is, videos for this week’s talks are already online.

As mentioned in the previous two posts, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1} \in {\bf Z}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

There is a higher dimensional generalisation, which first appeared explicitly (in a more general form) in this preprint of Szegedy (which used a slightly different argument than the one of Ben, Tammy, and myself; see also this previous preprint of Szegedy with related results):

Theorem 2 (Inverse theorem for multidimensional Gowers norms) Let {N \geq 1} and {s,d \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z}^d \rightarrow [-1,1]} is a function supported on {[N]^d} such that

\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n,h_1,\dots,h_{s+1} \in {\bf Z}^d} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta. \ \ \ \ \ (1)

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta,d}(1)}, a polynomial sequence {g: {\bf Z}^d \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta,d}(1)} such that

\displaystyle \frac{1}{N^d} \sum_{n \in {\bf Z}^d} f(n) F(g(n) \Gamma) \gg_{s,\delta,d} 1.

The {d=2} case of this theorem was recently used by Wenbo Sun. One can replace the polynomial sequence with a linear sequence if desired by using a lifting trick (essentially due to Furstenberg, but which appears explicitly in Appendix C of my paper with Ben and Tammy).

In this post I would like to record a very neat and simple observation of Ben Green and Nikos Frantzikinakis, that uses the tool of Freiman isomorphisms to derive Theorem 2 as a corollary of the one-dimensional theorem. Namely, consider the linear map {\phi: {\bf Z}^d \rightarrow {\bf Z}} defined by

\displaystyle \phi( n_1,\dots,n_d ) := \sum_{i=1}^d (10 N)^{i-1} n_i,

that is to say {\phi} is the digit string base {10N} that has digits {n_d \dots n_1}. This map is a linear map from {[N]^d} to a subset of {[d 10^d N^d]} of density {1/(d10^d)}. Furthermore it has the following “Freiman isomorphism” property: if {n, h_1,\dots,h_{s+1}} lie in {{\bf Z}} with {n + \omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} in the image set {\phi( [N]^d )} of {[N]^d} for all {\omega}, then there exist (unique) lifts {\tilde n \in {\bf Z}^d, \tilde h_1,\dots,\tilde h_{s+1} \in {\bf Z}} such that

\displaystyle \tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1} \in [N]^d


\displaystyle \phi( \tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1} ) = n + \omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}

for all {\omega}. Indeed, the injectivity of {\phi} on {[N]^d} uniquely determines the sum {\tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1}} for each {\omega}, and one can use base {10N} arithmetic to verify that the alternating sum of these sums on any {2}-facet of the cube {\{0,1\}^{s+1}} vanishes, which gives the claim. (In the language of additive combinatorics, the point is that {\phi} is a Freiman isomorphism of order (say) {8} on {[N]^d}.)

Now let {\tilde f: {\bf Z} \rightarrow [-1,1]} be the function defined by setting {\tilde f( \phi(n) ) := f(n)} whenever {n \in [N]^d}, with {\tilde f} vanishing outside of {\phi([N]^d)}. If {f} obeys (1), then from the above Freiman isomorphism property we have

\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n, h_1,\dots,h_{s+1} \in {\bf Z}} \prod_{\omega \in \{0,1\}^{s+1}} \tilde f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Applying the one-dimensional inverse theorem (Theorem 1), with {\delta} reduced by a factor of {d 10^d} and {N} replaced by {d 10^d N^d}, this implies the existence of a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta,d}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta,d}(1)} such that

\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n \in {\bf Z}} \tilde f(n) F(g(n) \Gamma) \gg_{s,\delta,d} 1

which by the Freiman isomorphism property again implies that

\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n \in {\bf Z}^d} f(n) F(g(\phi(n)) \Gamma) \gg_{s,\delta,d} 1.

But the map {n \mapsto g(\phi(n))} is clearly a polynomial map from {{\bf Z}^d} to {G} (the composition of two polynomial maps is polynomial, see e.g. Appendix B of my paper with Ben and Tammy), and the claim follows.

Remark 3 This trick appears to be largely restricted to the case of boundedly generated groups such as {{\bf Z}^d}; I do not see any easy way to deduce an inverse theorem for, say, {\bigcup_{n=1}^\infty {\mathbb F}_p^n} from the {{\bf Z}}-inverse theorem by this method.

Remark 4 By combining this argument with the one in the previous post, one can obtain a weak ergodic inverse theorem for {{\bf Z}^d}-actions. Interestingly, the Freiman isomorphism argument appears to be difficult to implement directly in the ergodic category; in particular, there does not appear to be an obvious direct way to derive the Host-Kra inverse theorem for {{\bf Z}^d} actions (a result first obtained in the PhD thesis of Griesmer) from the counterpart for {{\bf Z}} actions.

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let {s \geq 1} be an integer, and let {(X, T)} be an ergodic, countably generated measure-preserving system. Suppose that one has

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} f(T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}x)\ d\mu(x)

\displaystyle > 0

for all non-zero {f \in L^\infty(X)} (all {L^p} spaces are real-valued in this post). Then {(X,T)} is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree {\leq s} nilsystems, that is to say systems of the form {(G/\Gamma, x \mapsto gx)} for some degree {\leq s} filtered nilmanifold {G/\Gamma} and a group element {g \in G} that acts ergodically on {G/\Gamma}.

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of {{\bf Z}}-actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let {s \geq 1} be an integer, and let {(X, T)} be an ergodic, countably generated measure-preserving system. Suppose that one has

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} f\ d\mu

\displaystyle > 0

for all non-zero {f \in L^\infty(X)}, where {T^h f := f \circ T^h}. Then {(X,T)} is a factor of an inverse limit of ergodic degree {\leq s} nilsystems.

Theorem 4 (Pro-nilsystems closed under factors) Let {s \geq 1} be an integer. Then any factor of an inverse limit of ergodic degree {\leq s} nilsystems, is again an inverse limit of ergodic degree {\leq s} nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

Proposition 5 Theorem 1 implies Theorem 3.

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

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A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:

Theorem 1 (Discrete inverse theorem for Gowers norms) Let {N \geq 1} and {s \geq 1} be integers, and let {\delta > 0}. Suppose that {f: {\bf Z} \rightarrow [-1,1]} is a function supported on {[N] := \{1,\dots,N\}} such that

\displaystyle  \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a polynomial sequence {g: {\bf Z} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.

For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.

The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:

Theorem 2 (Continuous inverse theorem for Gowers norms) Let {s \geq 1} be an integer, and let {\delta>0}. Suppose that {f: {\bf R} \rightarrow [-1,1]} is a measurable function supported on {[0,1]} such that

\displaystyle  \int_{{\bf R}^{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(t+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1})\ dt dh_1 \dots dh_{s+1} \geq \delta. \ \ \ \ \ (1)

Then there exists a filtered nilmanifold {G/\Gamma} of degree {\leq s} and complexity {O_{s,\delta}(1)}, a (smooth) polynomial sequence {g: {\bf R} \rightarrow G}, and a Lipschitz function {F: G/\Gamma \rightarrow {\bf R}} of Lipschitz constant {O_{s,\delta}(1)} such that

\displaystyle  \int_{\bf R} f(t) F(g(t) \Gamma)\ dt \gg_{s,\delta} 1.

The interval {[0,1]} can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of {f}. Note though that the coefficients of {g} can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form {f(t) = \cos( \xi t)} for some arbitrarily large frequency {\xi}).

It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of {{\bf Z}} with {{\bf R}} (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval {[0,1]} as a limit of the discrete interval {\frac{1}{N} \cdot [N]} as {N \rightarrow \infty}. However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence {g: {\bf N} \rightarrow G} produced by Theorem 1, after normalising these coefficients by {N}. Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting {g} into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of {f}.

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Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:

Theorem 1 (Szemerédi’s theorem) Let {N} be a positive integer, and let {f: {\bf Z}/N{\bf Z} \rightarrow [0,1]} be a function with {{\bf E}_{x \in {\bf Z}/N{\bf Z}} f(x) \geq \delta} for some {\delta>0}, where we use the averaging notation {{\bf E}_{x \in A} f(x) := \frac{1}{|A|} \sum_{x \in A} f(x)}, {{\bf E}_{x,r \in A} f(x) := \frac{1}{|A|^2} \sum_{x, r \in A} f(x)}, etc.. Then for {k \geq 3} we have

\displaystyle  {\bf E}_{x,r \in {\bf Z}/N{\bf Z}} f(x) f(x+r) \dots f(x+(k-1)r) \geq c(k,\delta)

for some {c(k,\delta)>0} depending only on {k,\delta}.

The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases {k=1,2} as they are trivial and somewhat degenerate.

There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.

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In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:

  • Every positive integer {m} has a prime factorisation

    \displaystyle  m = p_1 p_2 \dots p_r

    into (not necessarily distinct) primes {p_1,\dots,p_r}, which is unique up to rearrangement. Taking logarithms, we obtain a partition

    \displaystyle  \log m = \log p_1 + \log p_2 + \dots + \log p_r

    of {\log m}.

  • (Prime number theorem) A randomly selected integer {m} of size {m \sim N} will be prime with probability {\approx \frac{1}{\log N}} when {N} is large.
  • If {m \sim N} is a randomly selected large integer of size {N}, and {p = p_i} is a randomly selected prime factor of {m = p_1 \dots p_r} (with each index {i} being chosen with probability {\frac{\log p_i}{\log m}}), then {\log p_i} is approximately uniformly distributed between {0} and {\log N}. (See Proposition 9 of this previous blog post.)
  • The set of real numbers {\{ \frac{\log p_i}{\log m}: i=1,\dots,r \}} arising from the prime factorisation {m = p_1 \dots p_r} of a large random number {m \sim N} converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit {N \rightarrow \infty}. (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)

Now for the facts about the cycle decomposition of large permutations:

  • Every permutation {\sigma \in S_n} has a cycle decomposition

    \displaystyle  \sigma = C_1 \dots C_r

    into disjoint cycles {C_1,\dots,C_r}, which is unique up to rearrangement, and where we count each fixed point of {\sigma} as a cycle of length {1}. If {|C_i|} is the length of the cycle {C_i}, we obtain a partition

    \displaystyle  n = |C_1| + \dots + |C_r|

    of {n}.

  • (Prime number theorem for permutations) A randomly selected permutation of {S_n} will be an {n}-cycle with probability exactly {1/n}. (This was noted in this previous blog post.)
  • If {\sigma} is a random permutation in {S_n}, and {C_i} is a randomly selected cycle of {\sigma} (with each {i} being selected with probability {|C_i|/n}), then {|C_i|} is exactly uniformly distributed on {\{1,\dots,n\}}. (See Proposition 8 of this blog post.)
  • The set of real numbers {\{ \frac{|C_i|}{n} \}} arising from the cycle decomposition {\sigma = C_1 \dots C_r} of a random permutation {\sigma \in S_n} converges (in a suitable sense) to the Poisson-Dirichlet process in the limit {n \rightarrow \infty}. (Again, see this previous blog post for details.)

See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.

There is however something unsatisfying about the analogy, in that it is not clear why there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials {P \in {\mathbf F}_q[T]} over a finite field {{\mathbf F}_q}, discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials {P}. We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:

  • Every monic polynomial {f \in {\mathbf F}_q[T]} has a factorisation

    \displaystyle  f = P_1 \dots P_r

    into irreducible monic polynomials {P_1,\dots,P_r \in {\mathbf F}_q[T]}, which is unique up to rearrangement. Taking degrees, we obtain a partition

    \displaystyle  \hbox{deg} f = \hbox{deg} P_1 + \dots + \hbox{deg} P_r

    of {\hbox{deg} f}.

  • (Prime number theorem for polynomials) A randomly selected monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be irreducible with probability {\approx \frac{1}{n}} when {q} is fixed and {n} is large.
  • If {f \in {\mathbf F}_q[T]} is a random monic polynomial of degree {n}, and {P_i} is a random irreducible factor of {f = P_1 \dots P_r} (with each {i} selected with probability {\hbox{deg} P_i / n}), then {\hbox{deg} P_i} is approximately uniformly distributed in {\{1,\dots,n\}} when {q} is fixed and {n} is large.
  • The set of real numbers {\{ \hbox{deg} P_i / n \}} arising from the factorisation {f = P_1 \dots P_r} of a randomly selected polynomial {f \in {\mathbf F}_q[T]} of degree {n} converges (in a suitable sense) to the Poisson-Dirichlet process when {q} is fixed and {n} is large.

The above list of facts addressed the large {n} limit of the polynomial ring {{\mathbf F}_q[T]}, where the order {q} of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers {{\bf Z}}. However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large {q} limit where it is now the degree {n} that is held fixed, but the order {q} of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic {p} of the field goes to infinity (again keeping the degree {n} fixed), although all of the results proven below for the large {p} limit turn out to be true as well in the large {q} limit.

The large {q} (or large {p}) limit is technically a different limit than the large {n} limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large {q} limit:

Theorem 1 (Prime number theorem) The probability that a random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} is irreducible is {\frac{1}{n}+o(1)} in the limit where {n} is fixed and the characteristic {p} goes to infinity.

Proof: There are {q^n} monic polynomials {f \in {\mathbf F}_q[T]} of degree {n}. If {f} is irreducible, then the {n} zeroes of {f} are distinct and lie in the finite field {{\mathbf F}_{q^n}}, but do not lie in any proper subfield of that field. Conversely, every element {\alpha} of {{\mathbf F}_{q^n}} that does not lie in a proper subfield is the root of a unique monic polynomial in {{\mathbf F}_q[T]} of degree {f} (the minimal polynomial of {\alpha}). Since the union of all the proper subfields of {{\mathbf F}_{q^n}} has size {o(q^n)}, the total number of irreducible polynomials of degree {n} is thus {\frac{q^n - o(q^n)}{n}}, and the claim follows. \Box

Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula {\frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) q^d} for the number of irreducible monic polynomials of degree {n}.

Now we can give a precise connection between the cycle distribution of a random permutation, and (the large {p} limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:

Theorem 3 The partition {\{ \hbox{deg}(P_1), \dots, \hbox{deg}(P_r) \}} of a random monic polynomial {f= P_1 \dots P_r\in {\mathbf F}_q[T]} of degree {n} converges in distribution to the partition {\{ |C_1|, \dots, |C_r|\}} of a random permutation {\sigma = C_1 \dots C_r \in S_n} of length {n}, in the limit where {n} is fixed and the characteristic {p} goes to infinity.

We can quickly prove this theorem as follows. We first need a basic fact:

Lemma 4 (Most polynomials square-free in large {q} limit) A random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be square-free with probability {1-o(1)} when {n} is fixed and {q} (or {p}) goes to infinity. In a similar spirit, two randomly selected monic polynomials {f,g} of degree {n,m} will be coprime with probability {1-o(1)} if {n,m} are fixed and {q} or {p} goes to infinity.

Proof: For any polynomial {g} of degree {m}, the probability that {f} is divisible by {g^2} is at most {1/q^{2m}}. Summing over all polynomials of degree {1 \leq m \leq n/2}, and using the union bound, we see that the probability that {f} is not squarefree is at most {\sum_{1 \leq m \leq n/2} \frac{q^m}{q^{2m}} = o(1)}, giving the first claim. For the second, observe from the first claim (and the fact that {fg} has only a bounded number of factors) that {fg} is squarefree with probability {1-o(1)}, giving the claim. \Box

Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation {\sigma \in S_n} consisting of {c_k} cycles of length {k} for {k=1,\dots,r}, where {c_k} are nonnegative integers with {\sum_{k=1}^r k c_k = n}, is precisely

\displaystyle  \frac{1}{\prod_{k=1}^r c_k! k^{c_k}},

since there are {\prod_{k=1}^r c_k! k^{c_k}} ways to write a given tuple of cycles {C_1,\dots,C_r} in cycle notation in nondecreasing order of length, and {n!} ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree {n} that are the product of {c_k} irreducible polynomials of degree {k} is

\displaystyle  \frac{1}{\prod_{k=1}^r c_k!} \prod_{k=1}^r ( (\frac{1}{k}+o(1)) q^k )^{c_k} + o( q^n )

which simplifies to

\displaystyle  \frac{1+o(1)}{\prod_{k=1}^r c_k! k^{c_k}} q^n,

and the claim follows.

This was a fairly short calculation, but it still doesn’t quite explain why there is such a link between the cycle decomposition {\sigma = C_1 \dots C_r} of permutations and the factorisation {f = P_1 \dots P_r} of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in {S_n} with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on {S_n} is (extremely) non-abelian but invertible. Also, the multiplication of a degree {n} and a degree {m} polynomial is a degree {n+m} polynomial, whereas the group multiplication law on permutations does not take a permutation in {S_n} and a permutation in {S_m} and return a permutation in {S_{n+m}}.

I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.

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I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and its generalisations) provide a qualitatively precise description of sets of small doubling. The other question was to obtain a non-abelian analogue of inverse Littlewood-Offord theory.

Let me discuss the former question first. Gromov’s theorem tells us that if a finite subset {A} of a group {G} exhibits polynomial growth in the sense that {|A^n|} grows polynomially in {n}, then the group generated by {A} is virtually nilpotent (the converse direction also true, and is relatively easy to establish). This theorem has been strengthened a number of times over the years. For instance, a few years ago, I proved with Shalom that the condition that {|A^n|} grew polynomially in {n} could be replaced by {|A^n| \leq C n^d} for a single {n}, as long as {n} was sufficiently large depending on {C,d} (in fact we gave a fairly explicit quantitative bound on how large {n} needed to be). A little more recently, with Breuillard and Green, the condition {|A^n| \leq C n^d} was weakened to {|A^n| \leq n^d |A|}, that is to say it sufficed to have polynomial relative growth at a finite scale. In fact, the latter paper gave more information on {A} in this case, roughly speaking it showed (at least in the case when {A} was a symmetric neighbourhood of the identity) that {A^n} was “commensurate” with a very structured object known as a coset nilprogression. This can then be used to establish further control on {A}. For instance, it was recently shown by Breuillard and Tointon (again in the symmetric case) that if {|A^n| \leq n^d |A|} for a single {n} that was sufficiently large depending on {d}, then all the {A^{n'}} for {n' \geq n} have a doubling constant bounded by a bound {C_d} depending only on {d}, thus {|A^{2n'}| \leq C_d |A^{n'}|} for all {n' \geq n}.

In this paper we are able to refine this analysis a bit further; under the same hypotheses, we can show an estimate of the form

\displaystyle  \log |A^{n'}| = \log |A^n| + f( \log n' - \log n ) + O_d(1)

for all {n' \geq n} and some piecewise linear, continuous, non-decreasing function {f: [0,+\infty) \rightarrow [0,+\infty)} with {f(0)=0}, where the error {O_d(1)} is bounded by a constant depending only on {d}, and where {f} has at most {O_d(1)} pieces, each of which has a slope that is a natural number of size {O_d(1)}. To put it another way, the function {n' \mapsto |A^{n'}|} for {n' \geq n} behaves (up to multiplicative constants) like a piecewise polynomial function, where the degree of the function and number of pieces is bounded by a constant depending on {d}.

One could ask whether the function {f} has any convexity or concavity properties. It turns out that it can exhibit either convex or concave behaviour (or a combination of both). For instance, if {A} is contained in a large finite group, then {n \mapsto |A^n|} will eventually plateau to a constant, exhibiting concave behaviour. On the other hand, in nilpotent groups one can see convex behaviour; for instance, in the Heisenberg group {\begin{pmatrix}{} {1} {\mathbf Z} {\mathbf Z} \\ {0} {1} {\mathbf Z} \\ {0} {1} \end{pmatrix}}, if one sets {A} to be a set of matrices of the form {\begin{pmatrix} 1 & O(N) & O(N^3) \\ 0 & 1 & O(N) \\ 0 & 0 & 1 \end{pmatrix}} for some large {N} (abusing the {O()} notation somewhat), then {n \mapsto A^n} grows cubically for {n \leq N} but then grows quartically for {n > N}.

To prove this proposition, it turns out (after using a somewhat difficult inverse theorem proven previously by Breuillard, Green, and myself) that one has to analyse the volume growth {n \mapsto |P^n|} of nilprogressions {P}. In the “infinitely proper” case where there are no unexpected relations between the generators of the nilprogression, one can lift everything to a simply connected Lie group (where one can take logarithms and exploit the Baker-Campbell-Hausdorff formula heavily), eventually describing {P^n} with fair accuracy by a certain convex polytope with vertices depending polynomially on {n}, which implies that {|P^n|} depends polynomially on {n} up to constants. If one is not in the “infinitely proper” case, then at some point {n_0} the nilprogression {P^{n_0}} develops a “collision”, but then one can use this collision to show (after some work) that the dimension of the “Lie model” of {P^{n_0}} has dropped by at least one from the dimension of {P} (the notion of a Lie model being developed in the previously mentioned paper of Breuillard, Greenm, and myself), so that this sort of collision can only occur a bounded number of times, with essentially polynomial volume growth behaviour between these collisions.

The arguments also give a precise description of the location of a set {A} for which {A^n} grows polynomially in {n}. In the symmetric case, what ends up happening is that {A^n} becomes commensurate to a “coset nilprogression” {HP} of bounded rank and nilpotency class, whilst {A} is “virtually” contained in a scaled down version {HP^{1/n}} of that nilprogression. What “virtually” means is a little complicated; roughly speaking, it means that there is a set {X} of bounded cardinality such that {aXHP^{1/n} \approx XHP^{1/n}} for all {a \in A}. Conversely, if {A} is virtually contained in {HP^{1/n}}, then {A^n} is commensurate to {HP} (and more generally, {A^{mn}} is commensurate to {HP^m} for any natural number {m}), giving quite a (qualitatively) precise description of {A} in terms of coset nilprogressions.

The main tool used to prove these results is the structure theorem for approximate groups established by Breuillard, Green, and myself, which roughly speaking asserts that approximate groups are always commensurate with coset nilprogressions. A key additional trick is a pigeonholing argument of Sanders, which in this context is the assertion that if {A^n} is comparable to {A^{2n}}, then there is an {n'} between {n} and {2n} such that {A \cdot A^{n'}} is very close in size to {A^{n'}} (up to a relative error of {1/n}). It is this fact, together with the comparability of {A^{n'}} to a coset nilprogression {HP}, that allows us (after some combinatorial argument) to virtually place {A} inside {HP^{1/n}}.

Similar arguments apply when discussing iterated convolutions {\mu^{*n}} of (symmetric) probability measures on a (discrete) group {G}, rather than combinatorial powers {A^n} of a finite set. Here, the analogue of volume {A^n} is given by the negative power {\| \mu^{*n} \|_{\ell^2}^{-2}} of the {\ell^2} norm of {\mu^{*n}} (thought of as a non-negative function on {G} of total mass 1). One can also work with other norms here than {\ell^2}, but this norm has some minor technical conveniences (and other measures of the “spread” of {\mu^{*n}} end up being more or less equivalent for our purposes). There is an analogous structure theorem that asserts that if {\mu^{*n}} spreads at most polynomially in {n}, then {\mu^{*n}} is “commensurate” with the uniform probability distribution on a coset progression {HP}, and {\mu} itself is largely concentrated near {HP^{1/\sqrt{n}}}. The factor of {\sqrt{n}} here is the familiar scaling factor in random walks that arises for instance in the central limit theorem. The proof of (the precise version of) this statement proceeds similarly to the combinatorial case, using pigeonholing to locate a scale {n'} where {\mu *\mu^{n'}} has almost the same {\ell^2} norm as {\mu^{n'}}.

A special case of this theory occurs when {\mu} is the uniform probability measure on {n} elements {v_1,\dots,v_n} of {G} and their inverses. The probability measure {\mu^{*n}} is then the distribution of a random product {w_1 \dots w_n}, where each {w_i} is equal to one of {v_{j_i}} or its inverse {v_{j_i}^{-1}}, selected at random with {j_i} drawn uniformly from {\{1,\dots,n\}} with replacement. This is very close to the Littlewood-Offord situation of random products {u_1 \dots u_n} where each {u_i} is equal to {v_i} or {v_i^{-1}} selected independently at random (thus {j_i} is now fixed to equal {i} rather than being randomly drawn from {\{1,\dots,n\}}. In the case when {G} is abelian, it turns out that a little bit of Fourier analysis shows that these two random walks have “comparable” distributions in a certain {\ell^2} sense. As a consequence, the results in this paper can be used to recover an essentially optimal abelian inverse Littlewood-Offord theorem of Nguyen and Vu. In the nonabelian case, the only Littlewood-Offord theorem I am aware of is a recent result of Tiep and Vu for matrix groups, but in this case I do not know how to relate the above two random walks to each other, and so we can only obtain an analogue of the Tiep-Vu results for the symmetrised random walk {w_1 \dots w_n} instead of the ordered random walk {u_1 \dots u_n}.

Suppose that {A \subset B} are two subgroups of some ambient group {G}, with the index {K := [B:A]} of {A} in {B} being finite. Then {B} is the union of {K} left cosets of {A}, thus {B = SA} for some set {S \subset B} of cardinality {K}. The elements {s} of {S} are not entirely arbitrary with regards to {A}. For instance, if {A} is a normal subgroup of {B}, then for each {s \in S}, the conjugation map {g \mapsto s^{-1} g s} preserves {A}. In particular, if we write {A^s := s^{-1} A s} for the conjugate of {A} by {s}, then

\displaystyle  A = A^s.

Even if {A} is not normal in {B}, it turns out that the conjugation map {g \mapsto s^{-1} g s} approximately preserves {A}, if {K} is bounded. To quantify this, let us call two subgroups {A,B} {K}-commensurate for some {K \geq 1} if one has

\displaystyle  [A : A \cap B], [B : A \cap B] \leq K.

Proposition 1 Let {A \subset B} be groups, with finite index {K = [B:A]}. Then for every {s \in B}, the groups {A} and {A^s} are {K}-commensurate, in fact

\displaystyle  [A : A \cap A^s ] = [A^s : A \cap A^s ] \leq K.

Proof: One can partition {B} into {K} left translates {xA} of {A}, as well as {K} left translates {yA^s} of {A^s}. Combining the partitions, we see that {B} can be partitioned into at most {K^2} non-empty sets of the form {xA \cap yA^s}. Each of these sets is easily seen to be a left translate of the subgroup {A \cap A^s}, thus {[B: A \cap A^s] \leq K^2}. Since

\displaystyle [B: A \cap A^s] = [B:A] [A: A \cap A^s] = [B:A^s] [A^s: A \cap A^s]

and {[B:A] = [B:A^s]=K}, we obtain the claim. \Box

One can replace the inclusion {A \subset B} by commensurability, at the cost of some worsening of the constants:

Corollary 2 Let {A, B} be {K}-commensurate subgroups of {G}. Then for every {s \in B}, the groups {A} and {A^s} are {K^2}-commensurate.

Proof: Applying the previous proposition with {A} replaced by {A \cap B}, we see that for every {s \in B}, {A \cap B} and {(A \cap B)^s} are {K}-commensurate. Since {A \cap B} and {(A \cap B)^s} have index at most {K} in {A} and {A^s} respectively, the claim follows. \Box

It turns out that a similar phenomenon holds for the more general concept of an approximate group, and gives a “classification” of all the approximate groups {B} containing a given approximate group {A} as a “bounded index approximate subgroup”. Recall that a {K}-approximate group {A} in a group {G} for some {K \geq 1} is a symmetric subset of {G} containing the identity, such that the product set {A^2 := \{ a_1 a_2: a_1,a_2 \in A\}} can be covered by at most {K} left translates of {A} (and thus also {K} right translates, by the symmetry of {A}). For simplicity we will restrict attention to finite approximate groups {A} so that we can use their cardinality {A} as a measure of size. We call finite two approximate groups {A,B} {K}-commensurate if one has

\displaystyle  |A^2 \cap B^2| \geq \frac{1}{K} |A|, \frac{1}{K} |B|;

note that this is consistent with the previous notion of commensurability for genuine groups.

Theorem 3 Let {G} be a group, and let {K_1,K_2,K_3 \geq 1} be real numbers. Let {A} be a finite {K_1}-approximate group, and let {B} be a symmetric subset of {G} that contains {A}.

  • (i) If {B} is a {K_2}-approximate group with {|B| \leq K_3 |A|}, then one has {B \subset SA} for some set {S} of cardinality at most {K_1 K_2 K_3}. Furthermore, for each {s \in S}, the approximate groups {A} and {A^s} are {K_1 K_2^5 K_3}-commensurate.
  • (ii) Conversely, if {B \subset SA} for some set {S} of cardinality at most {K_3}, and {A} and {A^s} are {K_2}-commensurate for all {s \in S}, then {|B| \leq K_3 |A|}, and {B} is a {K_1^6 K_2 K_3^2}-approximate group.

Informally, the assertion that {B} is an approximate group containing {A} as a “bounded index approximate subgroup” is equivalent to {B} being covered by a bounded number of shifts {sA} of {A}, where {s} approximately normalises {A^2} in the sense that {A^2} and {(A^2)^s} have large intersection. Thus, to classify all such {B}, the problem essentially reduces to that of classifying those {s} that approximately normalise {A^2}.

To prove the theorem, we recall some standard lemmas from arithmetic combinatorics, which are the foundation stones of the “Ruzsa calculus” that we will use to establish our results:

Lemma 4 (Ruzsa covering lemma) If {A} and {B} are finite non-empty subsets of {G}, then one has {B \subset SAA^{-1}} for some set {S \subset B} with cardinality {|S| \leq \frac{|BA|}{|A|}}.

Proof: We take {S} to be a subset of {B} with the property that the translates {sA, s \in S} are disjoint in {BA}, and such that {S} is maximal with respect to set inclusion. The required properties of {S} are then easily verified. \Box

Lemma 5 (Ruzsa triangle inequality) If {A,B,C} are finite non-empty subsets of {G}, then

\displaystyle  |A C^{-1}| \leq |A B^{-1}| |B C^{-1}| / |B|.

Proof: If {ac^{-1}} is an element of {AC^{-1}} with {a \in A} and {c \in C}, then from the identity {ac^{-1} = (ab^{-1}) (bc^{-1})} we see that {ac^{-1}} can be written as the product of an element of {AB^{-1}} and an element of {BC^{-1}} in at least {|B|} distinct ways. The claim follows. \Box

Now we can prove (i). By the Ruzsa covering lemma, {B} can be covered by at most

\displaystyle \frac{|BA|}{|A|} \leq \frac{|B^2|}{|A|} \leq \frac{K_2 |B|}{|A|} \leq K_2 K_3

left-translates of {A^2}, and hence by at most {K_1 K_2 K_3} left-translates of {A}, thus {B \subset SA} for some {|S| \leq K_1 K_2 K_3}. Since {sA} only intersects {B} if {s \in BA}, we may assume that

\displaystyle  S \subset BA \subset B^2

and hence for any {s \in S}

\displaystyle  |A^s A| \leq |B^2 A B^2 A| \leq |B^6|

\displaystyle  \leq K_2^5 |B| \leq K_2^5 K_3 |A|.

By the Ruzsa covering lemma again, this implies that {A^s} can be covered by at most {K_2^5 K_3} left-translates of {A^2}, and hence by at most {K_1 K_2^5 K_3} left-translates of {A}. By the pigeonhole principle, there thus exists a group element {g} with

\displaystyle  |A^s \cap gA| \geq \frac{1}{K_1 K_2^5 K_3} |A|.


\displaystyle  |A^s \cap gA| \leq | (A^s \cap gA)^{-1} (A^s \cap gA)|


\displaystyle  (A^s \cap gA)^{-1} (A^s \cap gA) \subset A^2 \cap (A^s)^2

the claim follows.

Now we prove (ii). Clearly

\displaystyle  |B| \leq |S| |A| \leq K_3 |A|.

Now we control the size of {B^2 A}. We have

\displaystyle  |B^2 A| \leq |SA SA^2| \leq K_3^2 \sup_{s \in S} |A s A^2| = K_3^2 \sup_{s \in S} |A^s A^2|.

From the Ruzsa triangle inequality and symmetry we have

\displaystyle  |A^s A^2| \leq \frac{ |A^s (A^2 \cap (A^2)^s)| |(A^2 \cap (A^2)^s) A^2|}{|A^2 \cap (A^2)^s|}

\displaystyle  \leq \frac{ |(A^3)^s| |A^4| }{|A|/K_2}

\displaystyle  \leq K_2 \frac{ |A^3| |A^4| }{|A|}

\displaystyle  \leq K_1^5 K_2 |A|

and thus

\displaystyle  |B^2 A| \leq K_1^5 K_2 K_3^2 |A|.

By the Ruzsa covering lemma, this implies that {B^2} is covered by at most {K_1^5 K_2 K_3^2} left-translates of {A^2}, hence by at most {K_1^6 K_2 K_3^2} left-translates of {A}. Since {A \subset B}, the claim follows.

We now establish some auxiliary propositions about commensurability of approximate groups. The first claim is that commensurability is approximately transitive:

Proposition 6 Let {A} be a {K_1}-approximate group, {B} be a {K_2}-approximate group, and {C} be a {K_3}-approximate group. If {A} and {B} are {K_4}-commensurate, and {B} and {C} are {K_5}-commensurate, then {A} and {C} are {K_1^2 K_2^3 K_2^3 K_4 K_5 \max(K_1,K_3)}-commensurate.

Proof: From two applications of the Ruzsa triangle inequality we have

\displaystyle  |AC| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) (B^2 \cap C^2)| |(B^2 \cap C^2) C|}{|A^2 \cap B^2| |B^2 \cap C^2|}

\displaystyle  \leq \frac{|A^3| |B^4| |C^3|}{ (|A|/K_4) (|B|/K_5)}

\displaystyle  \leq K_4 K_5 \frac{K_1^2 |A| K_2^3 |B| K_3^2 |C|}{ |A| |B| }

\displaystyle  = K_1^2 K_2^3 K_3^2 K_4 K_5 |C|.

By the Ruzsa covering lemma, we may thus cover {A} by at most {K_1^2 K_2^3 K_3^2 K_4 K_5} left-translates of {C^2}, and hence by {K_1^2 K_2^3 K_3^3 K_4 K_5} left-translates of {C}. By the pigeonhole principle, there thus exists a group element {g} such that

\displaystyle  |A \cap gC| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|,

and so by arguing as in the proof of part (i) of the theorem we have

\displaystyle  |A^2 \cap C^2| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|

and similarly

\displaystyle  |A^2 \cap C^2| \geq \frac{1}{K_1^3 K_2^3 K_3^2 K_4 K_5} |C|

and the claim follows. \Box

The next proposition asserts that the union and (modified) intersection of two commensurate approximate groups is again an approximate group:

Proposition 7 Let {A} be a {K_1}-approximate group, {B} be a {K_2}-approximate group, and suppose that {A} and {B} are {K_3}-commensurate. Then {A \cup B} is a {K_1 + K_2 + K_1^2 K_2^4 K_3 + K_1^4 K_2^2 K_3}-approximate subgroup, and {A^2 \cap B^2} is a {K_1^6 K_2^3 K_3}-approximate subgroup.

Using this proposition, one may obtain a variant of the previous theorem where the containment {A \subset B} is replaced by commensurability; we leave the details to the interested reader.

Proof: We begin with {A \cup B}. Clearly {A \cup B} is symmetric and contains the identity. We have {(A \cup B)^2 = A^2 \cup AB \cup BA \cup B^2}. The set {A^2} is already covered by {K_1} left translates of {A}, and hence of {A \cup B}; similarly {B^2} is covered by {K_2} left translates of {A \cup B}. As for {AB}, we observe from the Ruzsa triangle inequality that

\displaystyle  |AB^2| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) B^2|}{|A^2 \cap B^2|}

\displaystyle  \leq \frac{|A^3| |B^4|}{|A|/K_3}

\displaystyle  \leq K_1^2 K_2^3 K_3 |B|

and hence by the Ruzsa covering lemma, {AB} is covered by at most {K_1^2 K_2^3 K_3} left translates of {B^2}, and hence by {K_1^2 K_2^4 K_3} left translates of {B}, and hence of {A \cup B}. Similarly {BA} is covered by at most {K_1^4 K_2^2 K_3} left translates of {B}. The claim follows.

Now we consider {A^2 \cap B^2}. Again, this is clearly symmetric and contains the identity. Repeating the previous arguments, we see that {A} is covered by at most {K_1^2 K_2^3 K_3} left-translates of {B}, and hence there exists a group element {g} with

\displaystyle  |A \cap gB| \geq \frac{1}{K_1^2 K_2^3 K_3} |A|.

Now observe that

\displaystyle  |(A^2 \cap B^2)^2 (A \cap gB)| \leq |A^5| \leq K_1^4 |A|

and so by the Ruzsa covering lemma, {(A^2 \cap B^2)^2} can be covered by at most {K_1^6 K_2^3 K_3} left-translates of {(A \cap gB) (A \cap gB)^{-1}}. But this latter set is (as observed previously) contained in {A^2 \cap B^2}, and the claim follows. \Box

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from this paper of Ben Green and myself) to obtain a slight amount of progress towards the open problem of obtaining {L^p} bounds for the trilinear and higher Hilbert transforms (as discussed in this previous blog post). For instance, the trilinear Hilbert transform

\displaystyle  H_3( f_1, f_2, f_3 )(x) := p.v. \int_{\bf R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}

is not known to be bounded for any {L^{p_1}({\bf R}) \times L^{p_2}({\bf R}) \times L^{p_3}({\bf R})} to {L^p({\bf R})}, although it is conjectured to do so when {1/p =1/p_1 +1/p_2+1/p_3} and {1 < p_1,p_2,p_3,p < \infty}. (For {p} well below {1}, one can use additive combinatorics constructions to demonstrate unboundedness; see this paper of Demeter.) One can approach this problem by considering the truncated trilinear Hilbert transforms

\displaystyle  H_{3,r,R}( f_1, f_2, f_3 )(x) := \int_{r \leq |t| \leq R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}

for {0 < r < R}. It is not difficult to show that the boundedness of {H_3} is equivalent to the boundedness of {H_{3,r,R}} with bounds that are uniform in {R} and {r}. On the other hand, from Minkowski’s inequality and Hölder’s inequality one can easily obtain the non-uniform bound of {2 \log \frac{R}{r}} for {H_{3,r,R}}. The main result of this paper is a slight improvement of this trivial bound to {o( \log \frac{R}{r})} as {R/r \rightarrow \infty}. Roughly speaking, the way this gain is established is as follows. First there are some standard time-frequency type reductions to reduce to the task of obtaining some non-trivial cancellation on a single “tree”. Using a “generalised von Neumann theorem”, we show that such cancellation will happen if (a discretised version of) one or more of the functions {f_1,f_2,f_3} (or a dual function {f_0} that it is convenient to test against) is small in the Gowers {U^3} norm. However, the arithmetic regularity lemma alluded to earlier allows one to represent an arbitrary function {f_i}, up to a small error, as the sum of such a “Gowers uniform” function, plus a structured function (or more precisely, an irrational virtual nilsequence). This effectively reduces the problem to that of establishing some cancellation in a single tree in the case when all functions {f_0,f_1,f_2,f_3} involved are irrational virtual nilsequences. At this point, the contribution of each component of the tree can be estimated using the “counting lemma” from my paper with Ben. The main term in the asymptotics is a certain integral over a nilmanifold, but because the kernel {\frac{dt}{t}} in the trilinear Hilbert transform is odd, it turns out that this integral vanishes, giving the required cancellation.

The same argument works for higher order Hilbert transforms (and one can also replace the coefficients in these transforms with other rational constants). However, because the quantitative bounds in the arithmetic regularity and counting lemmas are so poor, it does not seem likely that one can use these methods to remove the logarithmic growth in {R/r} entirely, and some additional ideas will be needed to resolve the full conjecture.