On the Uniform Distribution mod1 of Sequences
by
Stanislava Stoilova

Some new results are proved for uniform distribution of sequences and measures for the irregularity of the distribution of the sequences in [0, 1).

A measure is the so-called diaphony. The classical definition of the diaphony is based on using the trigonometric function system and it is defined by Zinterhof. In 1997 Hellekalek and Leeb defined dyadic version of the diaphony, the so-called dyadic diaphony, based on the Walsh functional system.

We will consider other two versions of the diaphony, based on the Chrestenson-Levy functional system and Price functional system. It is proved that the two versions of the diaphony are measures for uniform distribution and the exact order of the diaphony of one dimentional sequences is found; the so-called sequences of Faure.

A other results are shown in connection with criteria for uniform distribution. The so-called modified integrals of Price and modified integrals of Haar are defined their connection with uniform distribution mod1 is shown. Using the classic results, we obtain the analogues of LeVeque and Erdos-Turan inequalities in terms of this modified integrals.

Date received: July 25, 2001