A new upper bound for sets with no square differences

Abstract

We show that if $A\subset \{1,\dots ,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\ge 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemerédi.

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