Spectral gap in the group of affine transformations over prime fields

TL;DR

This paper investigates the spectral gap properties of random walks on affine groups over prime fields, linking the spectral gap of the entire group to that of its linear component, with applications to Euclidean isometries.

Contribution

It provides a new estimate of the spectral gap for affine transformations over prime fields based on the spectral gap of their linear parts.

Findings

Spectral gap of the affine group is estimated via the linear part.

Results have implications for the smoothness of self-similar measures.

Connections established between algebraic properties and geometric applications.

Abstract

We study random walks on the semi-direct product of F_p^d and SL_d(F_p). We estimate the spectral gap in terms of the spectral gap of the projection to the linear part SL_d(F_p). This problem is motivated by an analogue in the isometry group of Euclidean space, which have application to smoothness of self-similar measures.