A new upper bound for sets with no square differences. Issue 8 (30th August 2022)

• Record Type: Journal Article
• Authors: Bloom, Thomas F.; Maynard, James
• Is Part Of: Compositio mathematica. Volume 158:Issue 8(2022)
• Page Start: 1777
• Page End: 1798
• Publication Date: 2022-08-30
• Subjects: additive combinatorics -- squares -- difference set -- density increment; 11B30 -- 11P55 -- 11D09; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007679
• Languages: English
• ISSNs: 0010-437X

Abstract

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.