Memoirs of the American Mathematical Society 2009; 72 pp; softcover Volume: 201 ISBN10: 0821843249 ISBN13: 9780821843246 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/201/943
 Let \(f\) be a periodic measurable function and \((n_k)\) an increasing sequence of positive integers. The authors study conditions under which the series \(\sum_{k=1}^\infty c_k f(n_kx)\) converges in mean and for almost every \(x\). There is a wide classical literature on this problem going back to the 30's, but the results for general \(f\) are much less complete than in the trigonometric case \(f(x)=\sin x\). As it turns out, the convergence properties of \(\sum_{k=1}^\infty c_k f(n_kx)\) in the general case are determined by a delicate interplay between the coefficient sequence \((c_k)\), the analytic properties of \(f\) and the growth speed and numbertheoretic properties of \((n_k)\). In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the \(n_k\) are random and investigate the discrepancy the sequence \(\{n_kx\}\) mod 1. Table of Contents  Introduction
 Mean convergence
 Almost everywhere convergence: Sufficient conditions
 Almost everywhere convergence: Necessary conditions
 Random sequences
 Discrepancy of random sequences \(\{S_n x\}\)
 Some open problems
 Bibliography
