Gaps and the exponent of convergence of an integer sequence

TL;DR

This paper investigates how the gaps and ratios within an integer sequence affect its exponent of convergence, which relates to the sequence's density and growth rate.

Contribution

It extends previous questions by analyzing the influence of ratio sequences and gap sequences on the upper and lower exponential densities of integer sequences.

Findings

Gaps influence the exponent of convergence.

Ratios of consecutive terms affect the exponential densities.

New relationships between sequence structure and convergence properties.

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.