Lacunary sequences and permutations

Abstract

By a classical principle of analysis, sufficiently thin subsequences of general sequences of functions behave like sequences of independent random variables. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis. In contrast to "true" random processes, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change radically after rearrangement. The purpose of this paper is to survey some recent results of the authors on permuted function series. We will see that rearrangement properties of lacunary trigonometric series $\sum (a_k\cos n_kx+b_k \sin n_kx)$ and their nonharmonic analogues $\sum c_k f(n_kx)$ are intimately connected with the number theoretic properties of $(n_k)_{k \geq 1}$ and we will give a complete characterization of permutational invariance in terms of the Diophantine properties of $(n_k)_{k \geq 1}$. We will also see that in a certain statistical sense, permutational invariance is the "typical" behavior of lacunary sequences.