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- W4297142000 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 ${x in Z_N : 1_A*1_A(x) leq gamma N}$ or of the form ${x in Z_N : 1_A*1_A(x) < gamma N}$, where $A$ is a subset of $Z_N$. 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 $N^c$, $c = c(theta, eps,gamma) > 0$. 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." @default.
- W4297142000 created "2022-09-27" @default.
- W4297142000 creator A5001899130 @default.
- W4297142000 date "2011-08-07" @default.
- W4297142000 modified "2023-09-28" @default.
- W4297142000 title "Some properties of lower level-sets of convolutions" @default.
- W4297142000 doi "https://doi.org/10.48550/arxiv.1108.1578" @default.
- W4297142000 hasPublicationYear "2011" @default.
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