Weighted double character sums with Beatty sequences

Let [Formula: see text] be an irrational number and let [Formula: see text] be a real number. The corresponding Beatty sequence is defined as [Formula: see text] where [Formula: see text] is the largest integer not exceeding y. In this paper, we answer an open problem of Shparlinski. More specifically, bounds for weighted double character sums with Beatty sequences of the shape [Formula: see text] are obtained, where p is a prime, [Formula: see text] is a multiplicative character modulo p, [Formula: see text] is a positive integer, [Formula: see text] are subsets, [Formula: see text] and [Formula: see text] are complex numbers.