GCD sums from Poisson integrals and systems of dilated functions

Abstract

Abstract. Upper bounds for GCD sums of the form (gcd(n k , n ℓ )) 2α (n k n ℓ ) α are established, where (n k ) 1≤k≤N is any sequence of distinct positive integers and 0 < α ≤ 1; the estimate for α = 1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1/2. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson-Hunt-type inequality for systems of dilated functions of bounded variation or belonging to Lip 1/2 , a result that in turn settles two longstanding problems on the a.e. behavior of systems of dilated functions: the a.e. growth of sums of the form N k=1 f (n k x) and the a.e. convergence of ∞ k=1 c k f (n k x) when f is 1-periodic and of bounded variation or in Lip 1/2 .

GCD sums and complete sets of square-free numbers

“…The better bound exp(C log N log log N ) was found by Aistleitner, Berkes, and Seip in [1] by a method that also led to a full solution when 1/2 < α < 1 (see Section 7 for details). The research carried out in [1] made it clear that α = 1/2 represents a profound limiting case that deserves separate attention and requires novel techniques. This recognition led to the present investigation.…”

“…In spite of our efforts to obtain an optimal bound, there is still room for a slight improvement, as shown by the following result from [1]: For large N there exist integers n 1 < · · · < n N such that…”

“…gcd(n k , n ℓ ) n k n ℓ ≥ N exp c log N loglog N for an absolute constant c > 0 [1]. Thus it remains to decide whether the factor log log log N in the exponent on the right-hand side of (2) can be removed or replaced by a function growing even more slowly.…”

“…Programme. 1 The purpose of the present paper is to obtain the following new bound 1 for (1) when α = 1/2.…”

“…Gál proved that when α = 1, C N (log log N ) 2 is an optimal upper bound for (1), with C an absolute constant independent of N and the distinct positive integers n 1 , ..., n N . The problem solved by Gál had been posed by Koksma in the 1930s, based on the observation that such bounds would have implications for the uniform distribution of sequences (n k x) mod 1 for almost all x [10].…”