Patterns without a popular difference

Abstract

Which finite sets $P\subseteq {\mathbb{Z}}^r$ with $|P|\ge 3$ have the following property: for every $A\subseteq [N{]}^r$, there is some nonzero integer $d$ such that $A$ contains $({\alpha }^{|P|}-o(1))N^r$ translates of $d\cdot P=\{dp:p\in P\}$, where $\alpha =|A|/N^r$? Green showed that all 3-point $P\subseteq \mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P=\{a,a+b,a+c,a+b+c\}\subseteq \mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of translates of $d\cdot P$ that one can guarantee to find.

Computer code for appendix attached as ancillary files

References in corpus (1)

• A Szemeredi-type regularity lemma in abelian groups, with applications

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