%0 Journal Article
%T Metric discrepancy theory, functions of bounded variation and GCD sums
%A Christoph Aistleitner
%A Philipp A. Mayer
%A Volker Ziegler
%J Uniform Distribution Theory
%D 2010
%I Mathematical Institute of the Slovak Academy of Sciences
%X Let $f(x)$ be a $1$-periodic function of bounded variation having mean zero, and let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers. Then a result of Baker implies the upper bound $\left| \sum_{k=1}^N f(n_k x) \right| = \mathcal{O} ( \sqrt{N} (\log N)^{3/2+ \ve} )$ for almost all $x \in (0,1)$ in the sense of the Lebesgue measure. We show that the asymptotic order of $\left| \sum_{k=1}^N f(n_k x) \right|$ is closely connected with certain number-theoretic properties of the sequence $(n_k)_{k \geq 1}$, namely a certain function involving the greatest common divisor function. More exactly, we give an upper bound for the asymptotic order of $\left| \sum_{k=1}^N f(n_k x) \right|$ in terms of the function $$h_N (n_1, \dots,n_N) = \sum_{1 \leq k_1, k_2 \leq N} \frac{\gcd(n_{k_1},n_{k_2})}{\max(n_{k_1},n_{k_2})}.$$
%K Function of bounded variation
%K discrepancy
%K gcd sum.
%U http://www.boku.ac.at/MATH/udt/vol05/no1/7AisMaZie10-1.pdf