The largest support to escape chaos in random multiplicative functions.

Abstract

Let [Formula: see text] be a Steinhaus random multiplicative function and let [Formula: see text] be a finite set of integers. We show that convergence of [Formula: see text] to a standard complex normal distribution forces [Formula: see text]. We prove that this zero-density condition is sharp: For most sets [Formula: see text] of density [Formula: see text], and hence for some such sets, a complex Gaussian limit holds whenever [Formula: see text]. However, for positive density, the correct normalization is [Formula: see text] rather than [Formula: see text]. Thus, the additional factor [Formula: see text] is nontrivial precisely when the density does not tend to zero.