Metric discrepancy theory, functions of bounded variation and GCD sums

Let f(x) be a 1-periodic function of bounded variation having mean zero, and let (nk)k≥1 be an increasing sequence of positive integers. Then a result of Baker implies the upper bound ∣∣∣∑Nk=1 f(nkx)∣∣∣ = O (√N(logN)3/2+ε) for almost all x ∈ (0, 1) in the sense of the Lebesgue measure. We show that the asymptotic order of ∣∣∣∑Nk=1 f(nkx)∣∣∣ is closely connected with certain number-theoretic properties of the sequence (nk)k≥1, namely a certain function involving the greatest common divisor function. More exactly, we give an upper bound for the asymptotic order of ∣∣∣∑Nk=1 f(nkx)∣∣∣ in terms of the function hN (n1, . . . , nN ) = ∑ 1≤k1,k2≤N gcd(nk1 , nk2) max(nk1 , nk2) .