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ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

Part of: Sequences and sets Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  19 February 2016

TOMASZ SCHOEN  and
OLOF SISASK
TOMASZ SCHOEN
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland; schoen@amu.edu.pl
OLOF SISASK
Affiliation:
Department of Mathematics, KTH, 100 44 Stockholm, Sweden; sisask@kth.se

Abstract

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We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then

$$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$
where $c>0$ is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent $1/7$ cannot be replaced by any constant larger than $1/2$ . We also establish a related result, which says that sumsets $A+A+A$ contain long arithmetic progressions if $A\subset \{1,\ldots ,N\}$ , or high-dimensional affine subspaces if $A\subset \mathbb{F}_{q}^{n}$ , even if $A$ has density of the shape above.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B25: Arithmetic progressions 11K70: Harmonic analysis and almost periodicity
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e5
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
©The Author(s) 2016

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