## Abstract

Let *q* be a cube-free positive integer and $\chi \phantom{\rule{3.33333pt}{0ex}}\left(\text{mod}\phantom{\rule{3.33333pt}{0ex}}q\right)$ be a
non-principal Dirichlet character. Our main result is a Burgess-type
estimate for ${\sum}_{n\in A}\chi \left(n\right)$, where $A\subset [1,q]$ is the
union of *s* disjoint intervals ${I}_{1},...,{I}_{s}$. We obtain a
nontrivial estimate for the character sum over *A* whenever
$\left|A\right|{s}^{-1/2}>{q}^{1/4+\u03f5}$ and each interval *I _{j}* ($1\le j\le s$) has length $|{I}_{j}|>{q}^{\u03f5}$ for any $\u03f5>0$. 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].

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