## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Journal of Number Theory |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1992 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory