Growth in Some Finite Three-Dimensional Matrix Groups

2020/05/11 by Brendan Murphy, Murphy, Brendan, James Wheeler +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.2005.05077

20 pages

arxiv created 2020/05/11 · arxiv updated 2020/05/12

Abstract

We study the growth of product sets in some finite three-dimensional matrix groups. In particular, we prove two results about the group of 2× 2 upper triangular matrices over arbitrary finite fields: a product set estimate using techniques from multiplicative combinatorics, and an energy estimate using incidence geometry. The energy method gives better quantitative results, but only applies to small sets. We also prove an energy result for the Heisenberg group.

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