Gaps and the exponent of convergence of an integer sequence

Professor Tibor Šalát, 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 Šalát'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.

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