  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.
