Decoupling theorems are established that bound the measure of E_m ∩ E_n, showing the set system (E_n) has significantly less dependence than previously supposed and implying improved extra-divergence and slow-divergence variants of the Duffin-Schaeffer conjecture.
GCD sums and sum-product estimates
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abstract
In this note we prove a new estimate on so-called GCD sums (also called G\'{a}l sums), which, for certain coefficients, improves significantly over the general bound due to de la Bret\`{e}che 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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2019 1verdicts
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Decoupling theorems for the Duffin-Schaeffer problem
Decoupling theorems are established that bound the measure of E_m ∩ E_n, showing the set system (E_n) has significantly less dependence than previously supposed and implying improved extra-divergence and slow-divergence variants of the Duffin-Schaeffer conjecture.