On tripling constant of multiplicative subgroups

TL;DR

This paper establishes new bounds on the size of tripled multiplicative subgroups and their energy in prime fields, advancing understanding of their combinatorial properties.

Contribution

It proves a lower bound on the tripling of multiplicative subgroups and bounds the multiplicative energy of their nonzero shifts, providing new insights into their structure.

Findings

|3G| \,\gg\, |G|^2 / \log |G| for subgroups with |G| < p^{1/2}

Bound on multiplicative energy: E^*(G+x) \ll |G|^2 \log |G|

Results improve understanding of subgroup growth and energy in finite fields.

Abstract

We prove that any multiplicative subgroup G of the prime field f_p with |G| < p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log |G|, where x is an arbitrary nonzero residue.