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A new upper bound for sets with no square differences

Part of: Diophantine equations Additive number theory; partitions Sequences and sets

Published online by Cambridge University Press:  30 September 2022

Thomas F. Bloom  and
James Maynard
Thomas F. Bloom
Affiliation:
Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK bloom@maths.ox.ac.uk
James Maynard
Affiliation:
Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK james.alexander.maynard@gmail.com
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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.

Keywords

additive combinatoricssquaresdifference setdensity increment

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity 11P55: Applications of the Hardy-Littlewood method 11D09: Quadratic and bilinear equations
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 8 , August 2022 , pp. 1777 - 1798
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

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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