Rarified Sums of the Thue-morse Sequence

Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0 mod q, with an even binary digit sum and the corresponding number of k < n, k ≡ 0 mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S3,0(n) > 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q = 4N + 1. On the other hand, it is shown that the number of primes q ≤ x with this property is o(x/ log x). Finally, analoga for “higher parities” are provided.