Limit laws for cotangent and Diophantine sums
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910822548111360 |
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| author | Borda, Bence Frühwirth, Lorenz Hauke, Manuel |
| author_facet | Borda, Bence Frühwirth, Lorenz Hauke, Manuel |
| contents | 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 $\sum_{n=1}^N f(n α)/n^p$ in terms of a $1$-periodic function $f$ with a power singularity of order $p \ge 1$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_12085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Limit laws for cotangent and Diophantine sums Borda, Bence Frühwirth, Lorenz Hauke, Manuel Number Theory Dynamical Systems Probability 11J83, 11K50, 11L03, 37E10, 60F05 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 $\sum_{n=1}^N f(n α)/n^p$ in terms of a $1$-periodic function $f$ with a power singularity of order $p \ge 1$ 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. |
| title | Limit laws for cotangent and Diophantine sums |
| topic | Number Theory Dynamical Systems Probability 11J83, 11K50, 11L03, 37E10, 60F05 |
| url | https://arxiv.org/abs/2308.12085 |