Uniform distribution of two-term recurrence sequences

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Abstract

Let (FORMULA PRESENTED), A, B be rational integers and for (FORMULA PRESENTED). The sequence (un) is clearly periodic modulo m and we say that (un) is uniformly distributed modulo m if for every 5, every residue modulo m occurs the same number of times in the sequence of residues (FORMULA PRESENTED), where N is the period of (un) modulo m. If (un) is uniformly distributed modulo m then m divides N, so we write N=mf. Several authors have characterized those m for which (un) is uniformly distributed modulo m. In fact in this paper we will show that a much stronger property holds when m = pk, p a prime. Namely, if (un) is uniformly distributed modulo pk with period pkf, then every residue modulo pk appears exactly once in the sequence (FORMULA PRESENTED), for every s. We also characterize those composite m for which this more stringent property holds.

Original languageEnglish (US)
Pages (from-to)37-45
Number of pages9
JournalTransactions of the American Mathematical Society
Volume301
Issue number1
DOIs
StatePublished - May 1987

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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