A new upper bound for sets with no square differences

Abstract

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

Locations

• arXiv (Cornell University) - View - PDF
• Compositio Mathematica - View