Generalization of two results of the theory of uniform distribution

Abstract:

For a sequence ${x_1}, \ldots ,{x_N}$ of points in $[0,1]$ and a sequence ${p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)$ of nonnegative numbers, define the distribution function \[ g(x) = x - \sum \limits _{{x_k} < x} {{p_k}.} \] Let $\varphi$ be an increasing function on $[0,1]$ and $\varphi (0) = 0$. The main result of the paper is \[ F({D_N}) \leqslant \int _0^1 {\varphi \left ( {\left | {g(x)} \right |} \right )} dx \leqslant \varphi ({D_N}),\] where ${D_N}$ is the supremum norm of $g$ on $[0,1]$ and $F$ is the antiderivative of $\varphi$ with $F(0) = 0$. This result generalizes and improves an estimate of Niederreiter [1] for the ${L^2}$ discrepancy of the sequence ${x_1}, \ldots ,{x_N}$. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.