Illinois Journal of Mathematics

Diophantine equations and the LIL for the discrepancy of sublacunary sequences

Christoph Aistleitner

Abstract

Let $(n_k)_{k \geq1}$ be a lacunary sequence, i.e., a sequence of positive integers satisfying the Hadamard gap condition $n_{k+1}/ n_k \ge q >1, k \geq1$. By a classical result of Philipp (Acta Arith. 26 (1975) 241-251), the discrepancy $D_N$ of $(n_k x)_{k\ge1}$ mod 1 satisfies the law of the iterated logarithm, i.e., we have $1/(4 \sqrt{2}) \leq\limsup_{N \to\infty} N D_N(n_k x) (2 N \log\log N)^{-1/2} \leq C_q$ for almost all $x \in(0,1)$, where $C_q$ is a constant depending on $q$. Recently, Fukuyama computed the exact value of the $\limsup$ for $n_k=\theta^k$, where $\theta>1$, not necessarily an integer, and the author showed that for a large class of lacunary sequences the value of the $\limsup$ is the same as in the case of { i.i.d.} random variables. In the sublacunary case,\break the situation is much more complicated. Using methods of Berkes, Philipp and Tichy, we prove an exact law of the iterated logarithm for a large class of sublacunary growing sequences $(n_k)_{k \geq1}$, characterized in terms of the number of solutions of certain Diophantine equations, and show that the value of the $\limsup$ is the same as in the case of { i.i.d.} random variables.

Article information

Source
Illinois J. Math., Volume 53, Number 3 (2009), 785-815.

Dates
First available in Project Euclid: 4 October 2010

https://projecteuclid.org/euclid.ijm/1286212916

Digital Object Identifier
doi:10.1215/ijm/1286212916

Mathematical Reviews number (MathSciNet)
MR2727355

Zentralblatt MATH identifier
1258.11077

Citation

Aistleitner, Christoph. Diophantine equations and the LIL for the discrepancy of sublacunary sequences. Illinois J. Math. 53 (2009), no. 3, 785--815. doi:10.1215/ijm/1286212916. https://projecteuclid.org/euclid.ijm/1286212916

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