On the correlation of the Thue–Morse sequence

Let s q denote the sum of digits function in base q . The aim of this work is to estimate the exponential sums involving the sum of digits of shifted integers, namely of the form ∑ n ⩽ x e ( α 0 s q ( n ) + ⋯ + α k s q ( n + k ) ) , where k is a nonnegative integer and α 0 , … , α k are real numbers. We deduce from these estimates some results concerning the correlation of the Thue–Morse sequence.