Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence x ≡ yr (mod p); x, y ∈ N, x, y ≤ H, r ∈ U , for certain ranges of H and |U|, where U is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence x ≡ λ (mod p); x ∈ N, L < x < L+ p/n, is at most p 1 3 −c uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ Z[x] is a fixed polynomial without multiple roots in C, then the congruence xf(x) ≡ 1 (mod p), x ∈ N, x ≤ p, has at most p 1 3 −c solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ε and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ε.