On Bohr sets of integer valued traceless matrices

TL;DR

This paper proves that Bohr-zero non-periodic sets of traceless integer matrices intersect all conjugacy classes, implying their characteristic polynomials cover all possible polynomials of such matrices, using equidistribution results from advanced group theory.

Contribution

It establishes a non-trivial intersection property for Bohr-zero sets of traceless integer matrices and links this to the distribution of characteristic polynomials, extending understanding of algebraic and harmonic structures.

Findings

Bohr-zero non-periodic sets intersect all conjugacy classes

Characteristic polynomials of these sets are comprehensive

Uses equidistribution of random walks on finite-dimensional tori

Abstract

In this paper we show that any Bohr-zero non-periodic set $B$ of traceless integer valued matrices, denoted by $\Lambda$, intersects non-trivially the conjugacy class of any matrix from $\Lambda$. As a corollary, we obtain that the family of characteristic polynomials of $B$ contains all characteristic polynomials of matrices from $\Lambda$. The main ingredient used in this paper is an equidistribution result for an $SL_d(\mathbf{Z})$ random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work.