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arXiv · 1508.03814

Differences of subgroups in subgroups

Ilya D. Shkredov
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Abstract

We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and prove a "dual" sum-products estimate.

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Ilya D. Shkredov. 2015-08-16. Differences of subgroups in subgroups. https://arxiv.org/abs/1508.03814

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