Projection Theorems in the Presence of Expansions

TL;DR

This paper establishes a restricted projection theorem for a specific family of projections from higher-dimensional space to lower-dimensional space, relevant to problems in homogeneous dynamics and equidistribution.

Contribution

It introduces a new projection theorem for a one-dimensional family of projections, expanding the understanding of geometric measure theory in the context of homogeneous dynamics.

Findings

Proves a restricted projection theorem for a specific family of projections.

Connects the theorem to quantitative equidistribution problems.

Provides tools applicable to homogeneous dynamics studies.

Abstract

We prove a restricted projection theorem for a certain one dimensional family of projections from $\mathbb R^n$ to $\mathbb R^k$. The family we consider here arises naturally in the study of quantitative equidistribution problems in homogeneous dynamics.