The largest support to escape chaos in random multiplicative functions.

Authors

Max Wenqiang Xu

Published in

Proceedings of the National Academy of Sciences of the United States of America. Volume 123. Issue 38. Pages e2618812123. Sep 22, 2026. Epub Sep 14, 2026.

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.

PMID:
42735316
Bibliographic data and abstract were imported from PubMed on 15 Sep 2026.

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