A new upper bound for sets with no square differences

Thomas F. Bloom; James Maynard

A new upper bound for sets with no square differences

Number Theory 2021-02-25 v2 Combinatorics

Authors: Thomas F. Bloom , James Maynard

Abstract

We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then $|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}$ for some absolute constant $c>0$ . This improves upon a result of Pintz-Steiger-Szemer\'edi.

Keywords

bound arithmetic progressions multiplicative functions

Cite

@article{arxiv.2011.13266,
  title  = {A new upper bound for sets with no square differences},
  author = {Thomas F. Bloom and James Maynard},
  journal= {arXiv preprint arXiv:2011.13266},
  year   = {2021}
}

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A new upper bound for sets with no square differences

Abstract

Keywords

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