Gaps and the exponent of convergence of an integer sequence

Abstract

Professor Tibor \v{S}al\'at, at one of his seminars at Comenius University, Bratislava, asked to study the influence of gaps of an integer sequence A={a_1<a_2<...<a_n<...} on its exponent of convergence. The exponent of convergence of A coincides with its upper exponential density. In this paper we consider an extension of Professor \v{S}al\'at's question and we study the influence of the sequence of ratios a_m/a_{m+1} and of the sequence (a_{m+1}-a_m)/a_m on the upper and on the lower exponential densities of A.