Some properties of lower level-sets of convolutions

Abstract

In the present paper we prove a certain lemma about the structure of "lower level-sets of convolutions", which are sets of the form $\{x\in {\mathbb{Z}}_N:1_A\ast 1_A(x)\le \gamma N\}$ or of the form $\{x\in {\mathbb{Z}}_N:1_A\ast 1_A(x)<\gamma N\}$, where $A$ is a subset of ${\mathbb{Z}}_N$. One result we prove using this lemma is that if $|A|=\theta N$ and $|A+A| \leq (1-\eps) N$, $0 < \eps < 1$, then this level-set contains an arithmetic progression of length at least $N^c$, $c = c(θ, \eps,γ) > 0$. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with some ideas of Bourgain); however, our method of proof allows us to obtain non-tower-type quantitative dependence between the constant $c$ and the parameters $\theta$ and $\eps$. For various reasons (discussed in the paper) one might think, wrongly, that such results would only be possible for level-sets involving triple and higher convolutions.

20 pages; minor correction in statement of Theorems 3 and 4. Final pre-publication version

References in corpus (4)

• On the Bogolyubov-Ruzsa lemma
• Translation invariant equations and the method of Sanders
• Arithmetic progressions in sumsets and L^p-almost-periodicity
• Roth's theorem in many variables