Differences of subgroups in subgroups

TL;DR

This paper investigates the structure of subgroups in prime fields, establishing bounds on subset sizes and representations, and explores collinear triples and sum-product estimates in finite fields.

Contribution

It provides new bounds on subgroup subset sizes, non-representability results for subgroup sums, and analyzes collinear triples and sum-product phenomena in finite fields.

Findings

If A,G are subgroups with G smaller than p^{3/4} and A-A in G, then |A| is much smaller than |G|^{1/3}.

Large subgroups G with size less than p^{1/2-eps} cannot be expressed as G=A+B with A a subgroup and B arbitrary.

Derived bounds on the number of collinear triples in subsets of finite fields and established dual sum-product estimates.

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.