Abstract: I will discuss some recent joint work with Igor Shparlinski on character sums and its applications to some congruence equations. These have a lot of applications, for instance they give us some information about the distribution of quadratic residues. We will focus on the solvability of au=x modulo a prime p, where x lies in an interval I and u in some approximate subgroup U of Fp. It has been recently shown by Cilleruelo and Garaev that this equation is solvable for almost all a, provided that |I|> p5/8+ε and |U| > p3/8 (where U is a subgroup or a set of consecutive powers of a primitive root).
We will prove, for almost all primes, some bounds on character sums which allow us to obtain similar results for a wider range of sizes for U and I. The proof combines classical analytic number theory with combinatorics techniques.