Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914958403436544 |
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| author | Parkkonen, Jouni Paulin, Frédéric |
| author_facet | Parkkonen, Jouni Paulin, Frédéric |
| contents | We give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ between two divergent geodesics or a divergent geodesic and a compact locally convex subset in negatively curved locally symmetric spaces with exponentially mixing geodesic flow, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular orbifold recovering results of Sarnak, and to confirm and extend a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem Parkkonen, Jouni Paulin, Frédéric Differential Geometry Dynamical Systems Number Theory We give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ between two divergent geodesics or a divergent geodesic and a compact locally convex subset in negatively curved locally symmetric spaces with exponentially mixing geodesic flow, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular orbifold recovering results of Sarnak, and to confirm and extend a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields. |
| title | Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem |
| topic | Differential Geometry Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2409.18251 |