Affiliation:
1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University , Uniwersytetu Poznańskiego 4, 61-614 Poznań , Poland
Abstract
Abstract
We prove that every subset of $\{1,\dots , N\}$ that does not contain any solutions to a translation invariant equation $a_{1}x_{1}+\dots +a_{k}x_{k}=0$ with $k\ge 4$ has at most $$ \begin{align*} & \exp\big(-c(\log N)^{1/5+o(1)}\big)N\end{align*} $$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally, our method has the potential to yield an optimal estimate for this problem that matches Behrend’s classical lower bound. Our approach relies on a new result on almost-periodicity of convolutions.
Publisher
Oxford University Press (OUP)
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