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

Let F p be the field of residue classes modulo a large prime number p . We prove that if G is a subgroup of the multiplicative group F p * and if I ⊂ F p is an arithmetic progression, then | G ∩ I | = ( 1 + o ( 1 ) ) | G | I | / p + R , where | R | < ( | I | 1 / 2 + | G | 1 / 2 + | I | 1 / 2 | G | 3 / 8 p − 1 / 8 ) p o ( 1 ) . We use this bound to show that the number of solutions to the congruence x n ≡ 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).