Gaps and the exponent of convergence of an integer sequence

Professor Tibor alát, at one of his seminars at Comenius University, Bratislava, asked to study the influence of gaps of an integer sequence $A={a_1 < a_2 < \dots < a_n < \dots}$ 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 alát's question and we study the influence of the sequence of ratios $({\frac{a_m}{a_{m+1}}})_{m=1}^\infty}$ and of the sequence $({\frac{a_{m+1}-a_m}{a_{m}}})_{m=1}^\infty}$ on the upper and on the lower exponential densities of $A$.