Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 111.26801
Autor: Erdös, Pál; Taylor, S.J.
Title: On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences (In English)
Source: Proc. Lond. Math. Soc., III. Ser. 7, 598-615 (1957).
Review: The paper is devoted to the determination of the dimension of the sets of Lebesgues measure zero which are the only sets of ordinary and absolute convergence for the series (*) sumk = 1oo \sin(nk x+\muk) where 0 \leq \muk \leq 2 \pi and (nk) is an increasing sequence of integers satisfying the condition tk = nk+1 /nk \geq \rho > 1. Some of the theorems proved are:
(1) If tk is an integer for large values of k, and tk ––> oo, then sum |\sin nk x| < oo on a set of x's having the power of the continuum. (2) If nk = k! and 0 < y < \pi or \pi < y < 2 \pi, then the series sum |\sin(nk x-y)| < oo for no value of x. (3) If sum t-1k < oo, then sum |\sin(nk x-y)| < oo for every value of x in a set of power of continium. (4) If \lambda > 0, \mu > 0, \rho > 0 are constants such that \lambda kp for every integer k, then the dimensions (Besicovitch) of the set of x's for which sum |\sin(nk x-\muk)| < oo is zero if 0 < \rho < 1 and 1-1/\rho if \rho > 1. (5) If tk ––> oo then sum |\sin(nk x-\muk)| < oo in a set of values of x of dimension 1. If (nk) is an increasing sequence of integers, we denote the sets of values x for which ((nk x)), the fractional part of nk x, is not equidistributed in (0,1) by E. As an application the following theorems are proved: (6) E has zero Lebesgue measure. (7) There exists a finite constant C and an increasing sequence of integers (nk) such that nk+1-nk < C and such that E is not enumerable. (8) If (nk) is an increasing sequence of integers such that nk+1-nk < C, then E has dimension zero. (9) If nk < Ck\rho (k = 1,2,...) then E has dimension not greater than 1. – 1-1/\rho. (10) If tk \geq \rho > 1 then the set E of values has dimension 1.
Reviewer: J.A.Siddigi
Classif.: * 11K06 General theory of distribution modulo 1
42A55 Lacunary series
Index Words: approximation and series expansion
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