A Note on Olivier’s Theorem and Convergence in Erdos-Ulam Density

Abstract

Olivier’s Theorem says that if Σan is a convergent positive series and (a_n) is monotone decreasing, then na_n→0. Salat and Toma [4] proved that the monotonicity condition can be omitted if the convergence of (nan)n is replaced by the statistical convergence. The aim of this note is to give an alternative proof and generalization of this result.