Limit laws for cotangent and Diophantine sums

• Bence Borda ^[1] ; Lorenz Frühwirth ^[1] ; Manuel Hauke ^[1]
  1. [1] Graz University of Technology

     Graz University of Technology

     Graz, Austria

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 3, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01046-5
• Enlaces
  • Texto completo
• Resumen

  • Limit laws for ergodic averages with a power singularity over circle rotations were first proved by Sinai and Ulcigrai, as well as Dolgopyat and Fayad. In this paper, we prove limit laws with an estimate for the rate of convergence for the sum ΣN n=1 f (nα)/n p in terms of a 1-periodic function f with a power singularity of order p ≥ 1 at integers. Our results apply in particular to cotangent sums related to Dedekind sums, and to sums of reciprocals of fractional parts, which appear in multiplicative Diophantine approximation. The main tools are Schmidt’s method in metric Diophantine approximation, the Gauss–Kuzmin problem and the theory of ψ-mixing random variables.

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