Gaps between logs (Q2865921)

Let \(\xi_n\) denote the fractional part of \(\log_b n\), and \(\{\xi_{n,N}\}_{n=1,\dots,N}\) denote the ordered set of the first \(N\) elements of \(\xi_n\). For given \(N\), the gap distribution \(P_N(s)\) is defined as the probability density on \(\mathbb{R}_{\geq 0}\), NEWLINE\[NEWLINEP_N(s)=\frac{1}{N}\sum_{n=1}^N\delta(s-N(\xi_{n+1,N}-\xi_{n,N})).NEWLINE\]NEWLINE In the paper under review, the authors assume that \(b>1\) is a transcendental number, and prove validity of NEWLINE\[NEWLINE \lim_{N\to\infty}\int_0^\infty f(s)P_N(s)ds=\int_0^\infty f(s)P(s)ds, NEWLINE\]NEWLINE for any bounded continuous function \(f:\mathbb{R}_{\geq 0}\to\mathbb{R}\), where \(P(s)\) is determined explicitly in terms of logarithm function and derivatives of Pochhammer symbol. The presented technique also works for algebraic \(b\), although in general ends in more intricate limit distributions. By the way, the authors discuss the cases of an integer base \(b\) and \(r\)th root of an integer. Finally, they show that in the limit case \(b\to 1\), the distribution \(P(s)\) converges to the exponential distribution \(e^{-s}\).