Effective multi-equidistribution for translates of unipotent flows and Central limit theorems in inhomogeneous Diophantine approximation

TL;DR

This paper proves a central limit theorem for inhomogeneous Diophantine approximation with a fixed, non-Liouville shift, by establishing effective multi-equidistribution for flows on homogeneous spaces.

Contribution

It introduces an effective multi-equidistribution result for diagonal translates of unipotent flows, combining recent advances and height functions, to derive the CLT in this setting.

Findings

Proves a CLT for inhomogeneous Diophantine approximation with fixed shift.

Establishes effective multi-equidistribution for diagonal translates of unipotent flows.

Combines recent results and height functions to achieve the main theorem.

Abstract

In this paper, we prove a central limit theorem for inhomogeneous Diophantine approximation with a fixed shift, provided the shift is non-Liouville. This generalizes earlier work of Dolgopyat, Fayad, and Vinogradov~\cite{DFV}. This is achieved by translating the problem to one involving flows on homogeneous spaces. In this latter setting, we establish an effective multi-equidistribution result for diagonal translates of unipotent flows. This result is obtained by combining a recent result of Kim~\cite{Kim2024} with the height function construction of Shi~\cite{Shi20}. The central limit theorem is then deduced using the method of Bj\"orklund and Gorodnik~\cite{BG}.