Abstract
We prove a discretized Product Theorem for general simple Lie groups, in the spirit of Bourgain’s Discretized Sum-Product Theorem.
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References
Y. Benoist and N. de Saxcé. Spectral gap in compact simple Lie groups. preprint arXiv:1405.1808, 2014.
J. Bourgain. On the Erdős–Volkmann and Katz–Tao ring conjectures. GAFA, 13, 2003.
Bourgain J.: The discretized sum-product and projection theorems. Journal d’Analyse Mathématique. 112, 193–236 (2010)
J. Bourgain and A. Gamburd. On the spectral gap for finitely generated subgroups of SU(2). Inventiones Mathematicae, 171:83–121, 2008.
Bourgain J., Gamburd A.: A spectral gap theorem in SU(d). Journal of the European Mathematical Society, (5)14:1455–1511, 2012.
J. Bourgain and A. Yehudayoff. Expansion in \({SL(2,\mathbb{R})}\) and monotone expansion. GAFA, 23:1–41, 2013.
Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes. Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus. J. Amer. Math. Soc., (1)24:231–280, 2011.
Breuillard E., Green B.J., Tao T.: Approximate subgroups of linear groups. GAFA. 21, 774–819 (2011)
A. Eskin, S. Mozes, and H. Oh. On uniform exponential growth for linear groups. Invent. Math., (1)160:1–30, 2005.
H. A. Helfgott. Growth and generation in \({SL(2,\mathbb{Z}/p\mathbb{Z})}\). Annals of Mathematics, 167:601–623, 2008.
N.H. Katz and T. Tao Some connections between Falconer’s distance set conjecture and sets of Furstenburg type. New York Mathematical Journal, 7:149–187, 2001.
S. Łojasiewicz. Ensembles semi-analytiques. Notes from a course given in Orsay, 2006. available at https://perso.univ-rennes1.fr/michel.coste.
L. Pyber and E. Szabó. Growth in finite groups of Lie type of bounded rank. preprint arXiv:1005.1858, 2010.
N. de Saxcé. Borelian subgroups of simple Lie groups. preprint arXiv:1408.1579, 2014.
T.C. Tao. Product set estimates for non-commutative groups. Combinatorica, 28:547–594, 2008.
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The author is supported by ERC AdG Grant 267259.
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de Saxcé, N. A product theorem in simple Lie groups. Geom. Funct. Anal. 25, 915–941 (2015). https://doi.org/10.1007/s00039-015-0326-7
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DOI: https://doi.org/10.1007/s00039-015-0326-7