TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 2, Pages 609–642 S 0002-9947(99)02277-1 Article electronically published on August 10, 1999 RARIFIED SUMS OF THE THUE-MORSE SEQUENCE MICHAEL DRMOTA AND MARIUSZ SKA LBA Abstract. Let q be an odd number and S q,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 S 3,0 (n) > 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q =4 N + 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. 1. Introduction The Thue-Morse sequence [9], [5] is defined by t n =(−1) s(n) , (1) where s(n) denotes the number of ones in the binary representation of n. For any positive integer q and i ∈ Z we denote S q,i (n)=  0≤j<n,j≡i(mod q) t j . (2) In 1969 Newman [10] proved a remarkable conjecture of L. Moser saying that for any n ≥ 1 S 3,0 (n) > 0. More precisely, he proved that 3 α 20 < S 3,0 (n) n α < 5 · 3 α with α = log 3 log 4 . In 1983 Coquet [1] provided an explicit precise formula for S 3,0 (n) by the use of a continuous function ψ 3 (x) with period 1 which is nowhere differentiable (η 3 (n) ∈ {−1, 0, 1}): S 3,0 (n)= n log 3 log 4 · ψ 3  log n log 4  − η 3 (n) 3 . (3) Furthermore, he was able to identify min ψ([0, 1]) > 0 and max ψ([0, 1]). In general, (asymptotic) representations similar to (3) exist for any S q,i (n) (see [5] and section 2). But it is a non-trivial problem to decide whether the continuous function ψ q,i (x) has a zero or not. The only known examples where ψ q (x)= ψ q,0 (x) has no zero are q =3 k 5 l ([6]) and q = 17 ([7]). (Note that the assertion that ψ q,i (x) Received by the editors July 6, 1995 and, in revised form, December 2, 1997. 1991 Mathematics Subject Classification. Primary 11B85; Secondary 11A63. This work was supported by the Austrian Science Foundation, grant Nr. M 00233–MAT. c 1999 American Mathematical Society 609 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use