Mixing property and pseudo random sequences

We will give a summary about the relations between the spectra of the Perron--Frobenius operator and pseudo random sequences for 1-dimensional cases.Comment: Published at http://dx.doi.org/10.1214/074921706000000211 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Ergodic properties of {\beta}-adic Halton sequences

We investigate a parametric extension of the classical s-dimensional Halton sequence, where the bases are special Pisot numbers. In a one- dimensional setting the properties of such sequences have already been in- vestigated by several authors [5, 8, 23, 28]. We use methods from ergodic theory to in order to investigate the distribution behavior of multidimen- sional versions of such sequences. As a consequence it is shown that the Kakutani-Fibonacci transformation is uniquely ergodic

Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Ces\`aro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra

The dynamical point of view of low-discrepancy sequences

International audienceIn this overview we show by examples, how to associate certain sequences in the higher-dimensional unit cube to suitable dynamical systems. We present methods and notions from ergodic theory that serve as tools for the study of low-discrepancy sequences and discuss an important technique, cutting- and-stacking of intervals

Unexpected distribution phenomenon resulting from Cantor series expansions

We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.Comment: 32 page

On the metric theory of numbers in non-Archimedean settings

This thesis is a contribution to some fields of the metrical theory of numbers in non- Archimedean settings. This is a branch of number theory that studies and characterizes sets of numbers, which occur in a locally compact topological field endowed with a non- Archimedean absolute value. This is done from a probabilistic or measure-theoretic point of view. In particular, we develop new formulations of ergodicity and unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences. We use subsequence ergodic theory to establish a generalised metrical theory of continued fractions in both the settings of the p-adic numbers and the formal Laurent series over a finite field. We introduce the a-adic van der Corput sequence which significantly generalises the classical van der Corput sequence. We show that it provides a wealth of examples of low-discrepancy sequences which are very useful in the quasi-Monte Carlo method. We use our subsequential characterization of unique ergodicity to solve the generalised version of an open problem asked by O. Strauch on the distribution of the sequence of consecutive van der Corput sequences. In addition to these problems in ergodic methods and number theory, we employ some geometric measure theory to settle the positive characteristic analogue of an open problem asked by R.D. Mauldin on the complexity of the Liouville numbers in the field of formal Laurent series over a finite field by giving a complete characterization of all Hausdorff measures of the set of Liouville numbers