Gap series and functions of bounded variation (Q2495672)

This note is mainly concerned with an asymptotic behaviour of subsums of the series \(\sum f (xn_{k})\), where \((n_{k})\) is a strictly increasing sequence of integers, and \(f\) is a locally square integrable real measurable function on \(\mathbb{R}\) with period \(1\) satisfying \(\int_{0}^{1}f(x)dx=0\). The author proves that the above presented gap series does not behave like an independent random series when \(f\) is a function of bounded variation with rational discontinuity.