On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences. (English) Zbl 0111.26801

Erdős, Pál; Taylor, S. J.

Proc. Lond. Math. Soc., III. Ser. 7, 598-615 (1957).

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 (*) \(\sum_{k=1}^\infty \sin(n_k x+\mu_k)\) where \(0 \leq \mu_k \leq 2 \pi\) and \((n_k)\) is an increasing sequence of integers satisfying the condition \(t_k=n_{k+1} /n_k \geq \varrho >1\). Some of the theorems proved are:
(1) If \(t_k\) is an integer for large values of \(k\), and \(t_k \to \infty\), then \(\sum |\sin n_k x| < \infty\) on a set of \(x\)’s having the power of the continuum. (2) If \(n_k =k!\) and \(0<y<\pi\) or \(\pi < y < 2 \pi\), then the series \(\sum |\sin(n_k x-y)| < \infty\) for no value of \(x\). (3) If \(\sum t^{-1}_k < \infty\), then \(\sum |\sin(n_k x-y)| < \infty\) for every value of \(x\) in a set of power of continuum. (4) If \(\lambda >0\), \(\mu >0\), \(\varrho >0\) are constants such that \(\lambda k^p\) for every integer \(k\), then the dimensions (Besicovitch) of the set of \(x'\)s for which \(\sum |\sin(n_k x-\mu_k)| < \infty\) is zero if \(0 < \varrho <1\) and \(1-1/\varrho\) if \(\varrho>1\). (5) If \(t_k \to \infty\) then \(\sum |\sin(n_k x-\mu_k)| < \infty\) in a set of values of \(x\) of dimension 1. If \((n_k)\) is an increasing sequence of integers, we denote the sets of values \(x\) for which \(((n_k x))\), the fractional part of \(n_k x\), 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 \((n_k)\) such that \(n_{k+1}-n_k <C\) and such that \(E\) is not enumerable. (8) If \((n_k)\) is an increasing sequence of integers such that \(n_{k+1}-n_k<C\), then \(E\) has dimension zero. (9) If \(n_k<Ck^\varrho\) \((k=1,2,...)\) then \(E\) has dimension not greater than 1. — \(1-1/\varrho.\) (10) If \(t_k \geq \varrho > 1\) then the set \(E\) of values has dimension 1.

Reviewer: J.A.Siddigi

MSC:

11K06	General theory of distribution modulo \(1\)
42A55	Lacunary series of trigonometric and other functions; Riesz products

Keywords:

approximation and series expansion