Equidistribution with an error rate and Diophantine approximation over function fields

TL;DR

This paper establishes pointwise equidistribution with an explicit error rate for certain orbits in a function field setting and extends Diophantine approximation results to weighted inequalities, broadening understanding of distribution and solutions over function fields.

Contribution

It extends equidistribution results with error rates to a broader class of orbits and generalizes Diophantine approximation formulas to weighted inequalities over function fields.

Findings

Proves pointwise equidistribution with an error rate for $H$-orbits in $SL(d, extbf{K})/SL(d, extbf{Z})$.

Derives an asymptotic formula for solutions to weighted Diophantine inequalities.

Enables equidistribution results for unbounded functions of class $C_eta$.

Abstract

We prove pointwise equidistribution with an error rate of each $H$-orbit in $SL(d,\mathbf{K})/SL(d,\mathbf{Z})$ for a certain proper subgroup $H$ of horospherical group over a function field $\mathbf{K}$, extending a work of Kleinbock-Shi-Weiss. Moreover, we obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class $C_\alpha$, which was first introduced by Eskin-Margulis-Mozes.