Pointwise equidistribution for almost smooth functions with an error rate and Weighted L\'evy-Khintchin theorem

TL;DR

This paper advances the understanding of equidistribution and Diophantine approximation by establishing a pointwise equidistribution theorem with an explicit error rate for almost smooth functions and extending the Le9vy-Khintchin theorem to weighted best approximations.

Contribution

It proves a strengthened equidistribution theorem with an error rate and extends the Le9vy-Khintchin theorem to weighted approximations, building on recent homogeneous dynamics techniques.

Findings

Established a pointwise equidistribution theorem with an explicit error rate.

Extended the Le9vy-Khintchin theorem to weighted best approximations.

Utilized advanced techniques from homogeneous dynamics and recent research.

Abstract

The purpose of this article is twofold: to prove a pointwise equidistribution theorem with an error rate for almost smooth functions, which strengthens the main result of Kleinbock, Shi and Weiss (2017); and to obtain a L\'evy-Khintchin theorem for weighted best approximations, which extends the main theorem of Cheung and Chevallier (2019). To do so, we employ techniques from homogeneous dynamics and the methods developed in the work of Cheung-Chevallier (2019) and Shapira-Weiss (2022).