Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
Pratyush Sarkar

TL;DR
This paper proves effective equidistribution of large torus translates in certain arithmetic homogeneous spaces, leading to precise counting formulas for matrices with given characteristic polynomials, extending Eskin-Mozes-Shah results.
Contribution
It establishes effective equidistribution results for translates of tori in rank at most 2 groups, including SL(3,R), with applications to matrix counting problems.
Findings
Effective equidistribution for large torus translates in SL(3,R)/Γ.
Asymptotic counting formula with power saving error term.
Extension of Eskin-Mozes-Shah theorems to new settings.
Abstract
Let be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such of rank at most , in particular , we prove effective equidistribution of large translates of tori in . As an application, we obtain an asymptotic counting formula with a power saving error term for integral matrices with a specified characteristic polynomial. These effectivize celebrated theorems of Eskin-Mozes-Shah.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Operator Algebra Research · Markov Chains and Monte Carlo Methods
