On Distribution of Elements of Subgroups in Arithmetic Progressions Modulo a Prime

Abstract

Let \(\mathbb{F}_p\) be the field of residue classes modulo a large prime number p. We prove that if \(\mathcal{G}\) is a subgroup of the multiplicative group \(\mathbb{F}_p^*\) and if \(\mathcal{I} \subset \mathbb{F}_p\) is an arithmetic progression, then \(|\mathcal{G} \cap \mathcal{I}| = (1 + o(1))|\mathcal{G}|\mathcal{I}|/p + R\), where \(|R| < (|\mathcal{I}|^{1/2} + |\mathcal{G}|^{1/2} + |\mathcal{I}|^{1/2}|\mathcal{G}|^{3/8}p^{-1/8})p^{o(1)}\). We use this bound to show that the number of solutions to the congruence xⁿ ≡ λ (mod p), x ∈ ℕ, L < x < L + p/n, is at most p^{1/3−1/390+o(1)} uniformly over positive integers n, λ and L. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author (2013).