A new upper bound for sets with no square differences. Issue 8 (30th August 2022) - Bloom, Thomas F.; Maynard, James
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.