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A new upper bound for sets with no square differences

Published online by Cambridge University Press:  30 September 2022

Thomas F. Bloom
Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK
James Maynard
Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK


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.

Research Article
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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T.B. was supported by a postdoctoral grant funded by the Royal Society held at the University of Cambridge. J.M. was supported by a Royal Society Wolfson Merit Award, and funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 851318).


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