On Bohr sets of integer-valued traceless matrices

Fish, Alexander

doi:10.1090/proc/13743

On Bohr sets of integer-valued traceless matrices
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Proc. Amer. Math. Soc. 146 (2018), 625-636 Request permission

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 $SL_d(\mathbb {Z})$ random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

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