| Summary: | 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 $d
\cdot P = \{d p : p \in P\}$, 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.
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