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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
C. Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Mathematische Annalen 365 (2016), 473–496.
C. Aistleitner, I. Berkes and K. Seip, GCD sums from Poisson integrals and systems of dilated functions, Journal of the European Mathematical Society 17 (2015), 1517–1546.
C. Aistleitner, T. Lachmann and N. Technau, There is no Khintchine threshold for metric pair correlations, Mathematika 65 (2019), 929–949.
C. Aistleitner, G. Larcher and M. Lewko, Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems, Israel Journal of Mathematics 222 (2017), 463–485.
T. F. Bloom, S. Chow, A. Gafni and A. Walker, Additive energy and the metric Poisso-nian property, Mathematika 64 (2018), 679–700.
A. Bondarenko and K. Seip, GCD sums and complete sets of square-free numbers, Bulletin of the London Mathematical Society 47 (2015), 29–41.
A. Bondarenko and K. Seip, Large greatest common divisor sums and extreme values of the Riemann zeta function, Duke Mathematical Journal 166 (2017), 1685–1701.
R. de la Bretèche and G. Tenenbaum, Sommes de Gál et applications, Proceedings of the London Mathematical Society 119 (2019), 104–134.
T. Dyer and G. Harman, Sums involving common divisors, Journal of the London Mathematical Society 34 (1986), 1–11.
I. S. Gál, A theorem concerning Diophantine approximations, Nieuw Archief voor Wiskunde 23 (1949), 13–38.
M. Lewko and M. Radziwill, Refinements of Gál’s theorem and applications, Advances in Mathematics 305 (2017), 280–297.
B. Murphy, O. Roche-Newton and I. Shkredov, Variations on the sum-product problem, SIAM Journal on Discrete Mathematics 29 (2015), 514–540.
O. Roche-Newton and M. Rudnev, On the Minkowski distances and products of sum sets, Israel Journal of Mathematics 209 (2015), 507–526.
Z. Rudnick and P. Sarnak, The pair correlation function of fractional parts of polynomials, Communications in Mathematical Physics 194 (1998), 61–70.
T. Sanders, On the Bogolyubov-Ruzsa lemma, Analysis & PDE 5 (2012), 627–655.
A. Walker, The primes are not metric Poissonian, Mathematika 64 (2018), 230–236.
About this article
Cite this article
Bloom, T.F., Walker, A. GCD sums and sum-product estimates. Isr. J. Math. 235, 1–11 (2020). https://doi.org/10.1007/s11856-019-1932-0