Gaps between logs

TL;DR

This paper investigates the distribution of gaps between fractional parts of logarithms of positive integers, extending results to various bases and connecting to quantum graph statistics.

Contribution

It provides explicit formulas for the gap distribution of log n for arbitrary bases, including transcendental and algebraic cases, using Weyl equidistribution.

Findings

Gap distribution approaches exponential as base nears one

Results extend to logarithms with any base, including roots of integers

Explicit formulas derived for transcendental bases

Abstract

We calculate the limiting gap distribution for the fractional parts of log n, where n runs through all positive integers. By rescaling the sequence, the proof quickly reduces to an argument used by Barra and Gaspard in the context of level spacing statistics for quantum graphs. The key ingredient is Weyl equidistribution of irrational translations on multi-dimensional tori. Our results extend to logarithms with arbitrary base; we deduce explicit formulas when the base is transcendental or the r:th root of an integer. If the base is close to one, the gap distribution is close to the exponential distribution.