On certain character sums over smooth numbers

We give nontrivial bounds in various ranges for character sums of the form ∑n S(x,y) χ(R1(n))eq(R2(n)), where χ is a nonprincipal multiplicative character modulo a prime q, R1 and R2 are rational functions modulo q, and S(x,y) is the set of positive integers n ≤ x that are divisible only by primes p ≤ y. We also give sharper bounds in some special cases