Automorphic word maps and Amit--Ashurst conjecture

TL;DR

This paper investigates the Amit--Ashurst conjecture on probability distributions of words in finite nilpotent groups, providing improved bounds, settling some cases, and introducing new techniques for word distribution analysis.

Contribution

It extends previous results by improving bounds for class 2 nilpotent groups and introduces methods to find words with identical distributions on groups.

Findings

Improved bounds for probability distributions in finite nilpotent groups of class 2.

Settled the Amit--Ashurst conjecture in specific cases.

Developed elementary techniques for analyzing word distributions.

Abstract

In this article, we address Amit--Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of Camina, Iniguez, and Thillaisundaram by providing improved bounds for the case of finite nilpotent groups of class $2$ for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of Iniguez and Sangroniz using elementary techniques.