The distribution of dilating sets: a journey from Euclidean to hyperbolic geometry

TL;DR

This paper surveys the distributional properties of dilating sets in Euclidean and hyperbolic geometries, highlighting recent advances in Fourier decay and asymptotic analysis on hyperbolic surfaces.

Contribution

It generalizes known Euclidean results to hyperbolic settings and extends recent work on expanding circle arcs to unit tangent bundles.

Findings

Euclidean dilating sets exhibit specific Fourier decay properties.

Hyperbolic case shows asymptotic expansion for averages along expanding curves.

Extension of results to compact hyperbolic surfaces and tangent bundles.

Abstract

We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some generalizations, derived as a direct byproduct of recent developments on the problem of Fourier decay of finite measures. In the hyperbolic setting, we consider a natural upgrade of the problem to unit tangent bundles; confining ourselves to compact hyperbolic surfaces, we discuss an extension of our recent result with Ravotti on expanding circle arcs, establishing a precise asymptotic expansion for averages along expanding translates of homogeneous curves.