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 $(α^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p : p \in P\}$, where $α= |A|/N^r$? 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.