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

Abstract: A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah’s/Ratner’s theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar