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A new upper bound for sets with no square differences
- Part of
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 8 / August 2022
- Published online by Cambridge University Press:
- 30 September 2022, pp. 1777-1798
- Print publication:
- August 2022
-
- Article
- Export citation
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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
for some absolute constant
\[ \lvert \mathcal{A}\rvert \ll \frac{N}{(\log N)^{c\log\log \log N}} \]
$c>0$. This improves upon a result of Pintz, Steiger, and Szemerédi.
