On the law of the iterated logarithm for the discrepancy of lacunary sequences

@inproceedings{Aistleitner2008OnTL,
  title={On the law of the iterated logarithm for the discrepancy of lacunary sequences},
  author={Christoph Aistleitner},
  year={2008}
}
A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim supN→∞ NDN (nkx)(N log log N)−1/2 ≤ Cq a.e. The value of the limsup is a long standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for nk = θ k, θ > 1, not necessarily integer. We extend… CONTINUE READING

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