On the law of the iterated logarithm for the discrepancy of 〈nkx〉

نویسندگان

  • Christoph Aistleitner
  • István Berkes
  • Istvan Berkes
  • Robert F. Tichy
چکیده

By a well known result of Philipp (1975), the discrepancyDN (ω) of the sequence (nkω)k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (nk), for subexpenentially growing (nk), but in general the behavior of (nkω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (nk), in this paper we prove a law of the iterated logarithm for the discrepancy DN (ω) for subexponentially growing (nk) without number theoretic assumptions.

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تاریخ انتشار 2007