Gaps between logs. (18th August 2013)

• Record Type: Journal Article
• Authors: Marklof, Jens; Strömbergsson, Andreas
• Is Part Of: Bulletin of the London Mathematical Society. Volume 45:Part 6(2013:Dec.)
• Page Start: 1267
• Page End: 1280
• Publication Date: 2013-08-18
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms/bdt060
• Languages: English
• ISSNs: 0024-6093

Abstract

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, then the gap distribution is close to the exponential distribution.