Growth in Chevalley groups relatively to parabolic subgroups and some applications

2020/03/28 by Ilya D. Shkredov, Shkredov, Ilya D.
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2003.12785

openalex publication_date 2020/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a Chevalley group \mathbf G(q) and a parabolic subgroup P⊂ \mathbf G(q), we prove that for any set A there is a certain growth of A relatively to P, namely, either AP or PA is much larger than A. Also, we study a question about intersection of Aⁿ with parabolic subgroups P for large n. We apply our method to obtain some results on a modular form of Zaremba's conjecture from the theory of continued fractions and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than 1/2.

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