The fractal structure of rarefied sums of the Thue-Morse sequence

The p-rarefied subsequences of the well-known Thue-Morse sequence are those indexed by arithmetic progressions with difference p. We study the partial sums of such subsequences, for p an odd prime, by introducing fractal-like functions which exhibit strict self-similarity under scaling transformations and which approximate the partial sums within a controllable error, which we calculate explicitly for primes p satisfying a certain eigenvalue condition. The scaling properties of the approximating functions then determine the asymptotic growth of the partial sums; we obtain the growth rate explicitly for primes p such that the multiplicative order of 2 (mod p) is p - 1 or (p - 1) 2. We extend our results to a generalization of the Thue-Morse sequence which we define, for any b > 2, in terms of the base b representation of integers.