Gaps and the exponent of convergence of an integer sequence (Q2893507)

The authors study the influence of the size of the gaps of an integer sequence \(A=\{a_1<a_2<a_3<\dots \}\) on its exponential convergence introduced by \textit{G. Pólya} and \textit{G. Szegő} [Classics in Mathematics. Berlin: Springer (1998; Zbl 1053.00002)]. It can be proved that the exponent of convergence of \(A\) is given by the formula \(\limsup_{n\to \infty} (\log n)/(\log a_N)\). Related exponential density is defined by \(\lim_{n\to \infty} (\log (| A\cap [1,n]|))/(\log n)\). In the paper numerous examples, results and also problems indicating the influence of gaps on various number theoretical densities (asymptotic, exponential etc.) are presented. For instance, if \(g_n=a_{n+1}-a_n\) and \(\liminf_{n\to \infty} g_n/a_n>0\) then the exponential density of \(A\) vanishes. The arguments are elementary.