On the distribution of the residues of small multiplicative subgroups of $\mathbb{F}_p$

Abstract

Distributional properties of small multiplicative subgroups of $\mathbb{F}_p$ are obtained. In particular, it is shown that if H < $\mathbb{F}_p^*$ is of size larger than polylogarithmic in p, then, letting β < 1 be a fixed exponent, most elements of any coset aH (a ∈ $\mathbb{F}_p^*$, arbitrary) will not fall into the interval [−p ^β, p ^β] ∈ $\mathbb{F}_p$. The arguments are based on the theory of heights and results from additive combinatoric.