On some congruences and exponential sums

Let ε>0 be a fixed small constant, Fp be the finite field of p elements for prime p. We consider additive and multiplicative problems in Fp that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let M be an arbitrary subset of Fp. If #M>p1/3+ε and H⩾p2/3 or if #M>p3/5+ε and H⩾p3/5+ε then all, but O(p1−δ) elements of Fp can be represented in the form hm with h∈[1,H] and m∈M, where δ>0 depends only on ε. Furthermore, let X be an arbitrary interval of length H and s be a fixed positive integer. IfH>p17/35+ε,#M>p17/35+ε, then the number T6(λ) of solutions to the congruencem1x1s+m2x2s+m3x3s+m4x4s+m5x5s+m6x6s≡λmodp,mi∈M,xi∈X,i=1,…,6, satisfiesT6(λ)=H6(#M)6p(1+O(p−δ)).