Some applications of the functions of the system $\Gamma_{\mathcal{B}_s}$ to the theory of the uniformly distributed sequences

Abstract

The authors  use the $\mathcal{B}_s$-adic functions constructed in Cantor systems to show some of their applications to the theory of the uniformly distributed sequences. The notions of multidimensional  modified integrals from these  functions are introduced. The following  results - the LeVeque's inequality, the Koksma's formula, the Erdős-Turán-Koksma's inequality and the integral Weyl's criterion, are presented in the terms of the introduced integrals.