On Bohr sets of integer-valued traceless matrices

Alexander Fish

Abstract

In this paper we show that any Bohr-zero non-periodic set $B$ of traceless integer valued matrices, denoted by $\mathrm{\Lambda }$, intersects non-trivially the conjugacy class of any matrix from $\mathrm{\Lambda }$. As a corollary, we obtain that the family of characteristic polynomials of $B$ contains all characteristic polynomials of matrices from $\mathrm{\Lambda }$. The main ingredient used in this paper is an equidistribution result of Burgain–Furman–Lindenstrauss–Mozes.

Keywords: Ergodic Ramsey Theory, Measure Rigidity, Analytic Number Theory.

AMS Subject Classification: Primary: 37A45; Secondary: 11P99, 11C99.