Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method
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| Format: | Preprint |
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2024
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| _version_ | 1866912589896744960 |
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| author | Radziwiłł, Maksym Technau, Niclas |
| author_facet | Radziwiłł, Maksym Technau, Niclas |
| contents | The distribution of the properly renormalized gaps of $\sqrt{n} \,\mathrm{mod}\, 1$ with $n < N$ converges (when $N\rightarrow \infty$) to a non-standard limit distribution, as Elkies and McMullen proved in 2004 using techniques from homogeneous dynamics. In this paper we give an essentially self-contained proof based on the circle method. Our main innovation consists in showing that a new type of correlation functions of $\sqrt{n} \,\mathrm{mod}\, 1$ converge. To define these correlation functions we restrict, smoothly, to those $\sqrt{n} \,\mathrm{mod}\, 1$ that lie in minor arcs, i.e. away from rational numbers with small denominators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method Radziwiłł, Maksym Technau, Niclas Number Theory Dynamical Systems 11J71, 37A44, 65C20 The distribution of the properly renormalized gaps of $\sqrt{n} \,\mathrm{mod}\, 1$ with $n < N$ converges (when $N\rightarrow \infty$) to a non-standard limit distribution, as Elkies and McMullen proved in 2004 using techniques from homogeneous dynamics. In this paper we give an essentially self-contained proof based on the circle method. Our main innovation consists in showing that a new type of correlation functions of $\sqrt{n} \,\mathrm{mod}\, 1$ converge. To define these correlation functions we restrict, smoothly, to those $\sqrt{n} \,\mathrm{mod}\, 1$ that lie in minor arcs, i.e. away from rational numbers with small denominators. |
| title | Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method |
| topic | Number Theory Dynamical Systems 11J71, 37A44, 65C20 |
| url | https://arxiv.org/abs/2403.16493 |