On character sums and the exceptional set of a congruence problem

We prove an estimate of character sums. This bound and the method of solving multiplicative ternary problems are used to obtain new results about the cardinality of an exceptional set of a congruence problem modulo a prime p. In particular, we show that “almost all” residue classes modulo p are representable in the form xy(modp), where max{x,y}≪p58+ɛ.