Asymptotic equidistribution for partition statistics and topological invariants

TL;DR

This paper develops a general framework using a variant of Wright's Circle Method to establish asymptotic equidistribution, convexity, and log concavity of coefficients in generating functions across various mathematical contexts.

Contribution

It introduces a new approach based on Wright's Circle Method for proving asymptotic properties of coefficients in generating functions, applicable to diverse mathematical structures.

Findings

Proves asymptotic equidistribution for partition statistics.

Establishes results for Betti numbers of Hilbert schemes.

Analyzes cell counts in algebraic schemes.

Abstract

We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of G\"ottsche.