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GCD sums and sum-product estimates


In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretèche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.

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Correspondence to Aled Walker.

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Bloom, T.F., Walker, A. GCD sums and sum-product estimates. Isr. J. Math. 235, 1–11 (2020).

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