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 \((n_k)_{k \ge 1}\) be an increasing sequence of positive integers.
Then a result of Baker implies the upper bound
\[
\sum_{k=1}^{N} f(n_k x) = O\!\left(\sqrt{N} (\log N)^{3/2+\varepsilon}\right)
\]
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
\((n_k)_{k \ge 1}\), 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
\[
h_N(n_1, \ldots, n_N) = \sum_{1 \le k_1, k_2 \le N}
\frac{\gcd(n_{k_1}, n_{k_2})}{\max(n_{k_1}, n_{k_2})}.
\]