Sbornik: Mathematics 196:10 1495–1502 c©2005 RAS(DoM) and LMS Matematicheskĭı Sbornik 196:10 103–110 DOI 10.1070/SM2005v196n10ABEH003709 Uniform distribution and Voronŏı convergence

Abstract. There is a broad generalization of a uniformly distributed sequence according to Weyl where the frequency of elements of this sequence falling into an interval is defined by using a matrix summation method of a general form. In the present paper conditions for uniform distribution are found in the case where a regular Voronŏı method is chosen as the summation method. The proofs are based on estimates of trigonometric sums of a certain special type. It is shown that the sequence of the fractional parts of the logarithms of positive integers is not uniformly distributed for any choice of a regular Voronŏı method. Bibliography: 6 titles. § 1. Voronŏı means and types of uniform distribution Let q0> 0, qn 0. For a sequence sn, n = 0, 1,..., we set un = qns0 + qn−1s1 + · · ·+ q0sn q0 + q1 + · · ·+ qn. It is clear that if sj = s0, then un = s0. The sequence qn defines a Voronŏı summation method: if un → s as n→∞, then by definition sn → s (W, qn). The criterion of regularity of a W-method is qn q0 + · · ·+ qn → 0. (1.1) It turns out that any two regular Voronŏı methods are compatible, that is, if sn → s(W) and sn → s′(W ′), then s = s′. For q0 = q1 = · · · the Voronŏı method becomes the Cesàro method. If a W-method is regular and the sequence qn is non-decreasing, then (W, qn) includes the Cesàro method C: it follows from sn → s(C) that sn → s(W, qn). Condition