Capacity of sets and uniform distribution of sequences (Q1092101)

In this paper the following result is proved: Given a sequence of complex numbers \((c_ n)^{\infty}_{n=-\infty}\) and an increasing sequence of real numbers \((V(N))^{\infty}_{N=0}\) such that \(1\leq V(N)\leq N\). If W(n) is an increasing sequence of positive real numbers with \(\sum^{\infty}_{n=0}1/nW(n)<\infty\) and if \(\sum^{\infty}_{n=- \infty}| c_ n|^ 2(W(| n|)/(V(| n|))^ 2)<\infty\) then the set of points x such that \((1/V(N))\sum^{N}_{n=- N}c_ n e^{2\pi inx}\) is not bounded has generalized convex capacity zero relative to the sequence \(\lambda_ n=\sum^{\infty}_{k=n}1/kW(k).\) The definition of generalized convex capacity is taken from \textit{N. K. Bary} [A treatise on trigonometric series. I (Pergamon 1964; Zbl 0129.280), p. 403]. From this theorem the following results on uniform distribution mod 1 are derived: For any sequence \((x_ n)^{\infty}_{n=1}\) the set of y, such that \((x_ n+ny)^{\infty}_{n=1}\) is not uniformly distributed has dimension zero. - For a uniformly distributed sequence \((x_ n)\) the set of y, such that \((x_ n,ny)\) is not uniformly distributed has dimension zero. - For a strictly increasing sequence \(a_ n\) of positive integers with \(a_ n=O(n^ p)\) the set of y, such that \((a_ n\cdot y)\) is not uniformly distributed has dimension less than or equal to \(1- 1/p.\quad -\) For the Riesz mean (R, 1/n) and for any sequence \((x_ n)\) the set of y, such that \((x_ n+ny)\) is not (R, 1/n)-uniformly distributed has logarithmic capacity zero.