Growth of small generating sets in SL_n(ℤ/pℤ)

Gill, Nick and Helfgott, Harald Andres (2011). Growth of small generating sets in SL_n(ℤ/pℤ). International Mathematics Research Notices, 18 pp. 4226–4251.

DOI: https://doi.org/10.1093/imrn/rnq244

Abstract

Let G=SL_n and fix δ a positive number. We prove that there are positive numbers ε and C such that, for all fields K=ℤ/pℤ _{(p prime)}, and all sets A⊂G(K) that generate G(K), either |A|>p^^(n+1−δ), δ>0 or |A⋅A⋅A|≥C|A|^^(1+ε).

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