A characterization of limiting functions arising in mod-* convergence

TL;DR

This paper characterizes the limiting functions in mod-* convergence, providing new insights and examples, including applications to mod-Cauchy convergence related to Dedekind sums and modular geodesics.

Contribution

It offers a general characterization of limiting functions in mod-* convergence, extending to various probability distributions with non-vanishing Fourier transforms.

Findings

Characterization of limiting functions in mod-Gaussian convergence.

Extension of results to general mod-* convergence.

Introduction of new examples involving Dedekind sums and modular geodesics.

Abstract

In this note, we characterize the limiting functions in mod-Gausssian convergence; our approach sheds a new light on the nature of mod-Gaussian convergence as well. Our results in fact more generally apply to mod-* convergence, where * stands for any family of probability distributions whose Fourier transforms do not vanish. We moreover provide new examples, including two new examples of (restricted) mod-Cauchy convergence from arithmetics related to Dedekind sums and the linking number of modular geodesics.