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On Bohr sets of integer-valued traceless matrices


Author: Alexander Fish
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 37A45; Secondary 11P99, 11C99
DOI: https://doi.org/10.1090/proc/13743
Published electronically: August 30, 2017
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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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Additional Information

Alexander Fish
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006 Australia
Email: alexander.fish@sydney.edu.au

DOI: https://doi.org/10.1090/proc/13743
Keywords: Ergodic Ramsey theory, measure rigidity, analytic number theory
Received by editor(s): June 7, 2016
Received by editor(s) in revised form: June 20, 2016, and March 21, 2017
Published electronically: August 30, 2017
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society