On the pseudo-randomness of subsets related to primitive roots

Abstract

Many results have been proved on the distribution of the primitive roots. These results reflect certain random type properties of the set G p of the primitive roots modulo p. This fact motivates the question that in what extent behaves G p as a random subset of ℤ p ? First a much more general form of this problem is studied by using the notion of pseudo-randomness of subsets of ℤ n which has been introduced and studied recently by Dartyge and Sárközy. This is followed by the study of the pseudo-randomness of a subset of ℤ p defined by index properties. In both cases it turns out that these subsets possess strong pseudo-random properties (the well-distribution measure and correlation measure of order k are small) but the pseudo-randomness is not perfect: there is a pseudo-random measure (the symmetry measure) which is large.

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Correspondence to Cécile Dartyge.

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Research partially supported by the Hungarian National Foundation for Scientific Research, Grant no. K 67676, K 72731 and by French-Hungarian exchange program Balaton No. F-48/2006.

Research partially supported by the Hungarian National Foundation for Scientific Research, Grant no. K 67676 and by French-Hungarian exchange program Balaton No. F-48/2006.

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Dartyge, C., Sárközy, A. & Szalay, M. On the pseudo-randomness of subsets related to primitive roots. Combinatorica 30, 139–162 (2010). https://doi.org/10.1007/s00493-010-2534-y

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Mathematics Subject Classification (2000)

  • 11K45
  • 11A07
  • 11L40