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arXiv · 2012.02078

Extremal problems for GCDs

Ben Green·Aled Walker
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Abstract

We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $δ|A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} δ^{-2 - \varepsilon} XY/D^2$. This is a new result even when $δ= 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.

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Ben Green, Aled Walker. 2020-12-16. Extremal problems for GCDs. https://doi.org/10.1017/s0963548321000092

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