## Illinois Journal of Mathematics

### Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n\sb k\omega)\bmod1$

#### Abstract

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences $(n_{k}\omega)$ mod $1$. Here $(n_{k})$ is a sequence of integers satisfying a sub-Hadamard growth condition and such that linear Diophantine equations in the variables $n_{k}$ do not have too many solutions. The proof depends on a martingale embedding of the empirical process; the number-theoretic structure of $(n_k)$ enters through the behavior of the square function of the martingale.

#### Article information

Source
Illinois J. Math., Volume 50, Number 1-4 (2006), 107-145.

Dates
First available in Project Euclid: 12 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258059472

Digital Object Identifier
doi:10.1215/ijm/1258059472

Mathematical Reviews number (MathSciNet)
MR2247826

Zentralblatt MATH identifier
1145.11058

#### Citation

Berkes, István; Philipp, Walter; Tichy, Robert F. Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n\sb k\omega)\bmod1$. Illinois J. Math. 50 (2006), no. 1-4, 107--145. doi:10.1215/ijm/1258059472. https://projecteuclid.org/euclid.ijm/1258059472