Steklov Institute of Mathematics at St.Petersburg

PREPRINT 15/2002

HARMONIC ANALYSIS ON TOTALLY DISCONNECTED GROUPS AND IRREGULARITIES OF POINT DISTRIBUTIONS

This preprint was accepted August 15, 2002
Contact: M. M. Skriganov

ABSTRACT:
The goal of this paper is to study point distributions
in the multi-dimensional unit cube which possess the structure
of finite abelian groups with respect to certain $p$-ary 
arithmetic operations. Such distributions can be thought
of as finite subgroups in a compact totally disconnected group
of the Cantor type. We apply the methods of $L^q$ harmonic
analysis to estimate very precisely the $L^q$-discrepancies for
such distributions. Following this approach, we explicitly
construct point distributions with the minimal order of the
$L^q$-discrepancy for each $q, $1\le q\le \infty$.