[Paper Review] A probabilistic technique for finding almost-periods of convolutions

Ernie Croot, Olof Sisask|arXiv (Cornell University)|Mar 15, 2010

Limits and Structures in Graph Theory29 references4 citations

TL;DR

This paper introduces a novel probabilistic method to find almost-periods in convolutions of subsets within arbitrary groups, bypassing the need for Fourier analysis. It establishes $L^2$-almost-periodicity results under local conditions, enabling new proofs of Roth's theorem and structured product set theorems in both abelian and non-abelian settings, particularly for sparse sets with small doubling or high multiplicative energy.

ABSTRACT

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.

Motivation & Objective

• To develop a new probabilistic method for identifying almost-periods in convolutions of subsets in arbitrary groups, avoiding reliance on Fourier analysis.
• To extend Bogolyubov-type and Freiman-type results to non-abelian and sparse settings where traditional Fourier-analytic tools fail.
• To provide a local version of almost-periodicity theorems that apply even when sets are not dense in their ambient group.
• To offer new, probabilistic proofs of classical results such as Roth’s theorem on three-term arithmetic progressions.
• To establish structural results for product sets $A_1 \cdot A_2 \cdot A_3$ and $A^2 \cdot A^{-2}$ under small-doubling or high-energy conditions.

Proposed method

• Introduces a probabilistic selection process to identify a large subset $T \subseteq S$ such that the convolution $1_A * 1_B$ is almost periodic under left-translation by elements of $TT^{-1}$.
• Uses moment bounds on hypergeometric and binomial distributions to control deviations in convolution values, leveraging Hoeffding’s inequality and moment generating function estimates.
• Applies concentration inequalities (e.g., Chernoff-type bounds) to derive tail estimates for sums of independent Bernoulli variables, enabling control over $L^2$-norm deviations.
• Employs a decomposition of the $L^2$-norm of the difference $\|1_A*1_B(xt) - 1_A*1_B(x)\|_2^2$ into tail integrals, bounded using exponential moment bounds.
• Reduces the problem of bounding $\mathbb{E}|X - np|^{2m}$ for binomial variables to gamma function and exponential integral estimates, yielding polynomial bounds in $n, p, m$.
• Establishes a comparison between hypergeometric and binomial moments via convex stochastic ordering, allowing transfer of known binomial moment bounds to hypergeometric settings.

Experimental results

Research questions

• RQ1 Can almost-periodicity of convolutions be established in arbitrary groups without relying on Fourier analysis?
• RQ2 To what extent can local conditions—such as $|B \cdot S| \leq K|B|$—replace global density assumptions in convolution structure theorems?
• RQ3 Can this probabilistic method yield new proofs of Roth’s theorem on three-term arithmetic progressions in the integers?
• RQ4 What structural properties emerge in product sets $A_1 \cdot A_2 \cdot A_3$ and $A^2 \cdot A^{-2}$ when $A$ has small doubling or high multiplicative energy?
• RQ5 How do these results extend to non-abelian groups, particularly in the context of strong approximate groups and sumset structures?

Key findings

• For any finite subsets $A, B \subseteq G$ and $\epsilon \in (0,1)$, there exists a set $T \subseteq S$ with $|T| \geq |S| / (2K)^{9/\epsilon^2}$ such that $\|1_A*1_B(xt) - 1_A*1_B(x)\|_2^2 \leq \epsilon^2 |A||B|^2$ for all $t \in TT^{-1}$, under the condition $|B \cdot S| \leq K|B|$.
• The method yields a probabilistic proof of Roth’s theorem on three-term arithmetic progressions in $\{1, \ldots, N\}$, valid for sets of density $\beta > 0$.
• A variant of the Bourgain-Green theorem on long arithmetic progressions in sumsets $A + B$ is proven to hold for sparser sets than previously known, under the same probabilistic framework.
• In non-abelian settings, the product sets $A_1 \cdot A_2 \cdot A_3$ and $A^2 \cdot A^{-2}$ contain very large iterated product sets when $A$ satisfies small-doubling or high multiplicative energy conditions.
• The results are local: they apply even when $A$ and $B$ are sparse, provided they interact well with some large set $S$, as formalized by the $|B \cdot S| \leq K|B|$ condition.
• The method extends to general functions via moment bounds on binomial and hypergeometric distributions, with $\mathbb{E}|X - np|^{2m} \leq 2(3mnp + m^2)^m$ for binomial variables, enabling precise control of convolution deviations.