On the law of the iterated logarithm for the discrepancy of Lacunary sequences.

By Christoph Aistleitner. 2010

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A classical result of Philipp (1975) states that for any sequence (n k ) k ≥1 of integers satisfying the Hadamard gap condition n k+1 /n k ≥ q > 1 (k = 1,2,...), the discrepancy D N of the sequence (n k x)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim sup ND N (n k x)(N log log N) -12 ≤ C q a .e. N→∞ The value of the lim sup is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for n k = θ k , θ > 1, not necessarily integer. We extend Fukuyama's result to a large class of integer sequences (n k ) characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

Subfields: Algebra and Number Theory, Information Systems, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/74f0f456836ff768caaa091d471ecea2d7471dac