Character sums over unions of intervals

Let q be a cube-free positive integer and χ(modq) be a non-principal Dirichlet character. Our main result is a Burgess-type estimate for σ n∈A χ(n), where A ⊂[1,q] is the union of s disjoint intervals I 1, ⋯,I s. We obtain a nontrivial estimate for the character sum over A whenever |A|s -1/2 >q 1/4+ε and each interval I_j (1≤j≤s) has length |I j |>q ε for any ε>0. This follows from an improvement of a mean value Burgess-type estimate studied by Heath-Brown [Number Theory and Related Fields, Springer Proc. Math. Statist. 43, New York (2013), 199-213].