Lacunary sequences in analysis, probability and number theory

In this paper we present the theory of lacunary trigonometric sums
and lacunary sums of dilated functions, from the origins of the subject up to re-
cent developments. We describe the connections with mathematical topics such
as equidistribution and discrepancy, metric number theory, normality, pseudo-
randomness, Diophantine equations, and the subsequence principle. In the final
section of the paper we prove new results which provide necessary and sufficient
conditions for the central limit theorem for subsequences, in the spirit of Nikishin’s
resonance theorem for convergence systems. More precisely, we characterize those
sequences of random variables which allow to extract a subsequence satisfying a
strong form of the central limit theorem.