Raney Transducers and the Lowest Point of the $p$-Lagrange spectrum
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909647163621376 |
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| author | Dong, Brandon Dupont, Soren O'Dorney, Evan M. Waitkus, W. Theo |
| author_facet | Dong, Brandon Dupont, Soren O'Dorney, Evan M. Waitkus, W. Theo |
| contents | It is well known that the golden ratio $ϕ$ is the ''most irrational'' number in the sense that its best rational approximations $s/t$ have error $\sim 1/(\sqrt{5} t^2)$ and this constant $\sqrt{5}$ is as low as possible. Given a prime $p$, how can we characterize the reals $x$ such that $x$ and $p x$ are both ''very irrational''? This is tantamount to finding the lowest point of the $p$-Lagrange spectrum $\mathcal{L}_p$ as previously defined by the third author. We describe an algorithm using Raney transducers that computes $\min \mathcal{L}_p$ if it terminates, which we conjecture it always does. We verify that $\min \mathcal{L}_p$ is the square root of a rational number for primes $p < 2000$. Mysteriously, the highest values of $\min \mathcal{L}_p$ occur for the Heegner primes $67$, $3$, and $163$, and for all $p$, the continued fractions of the corresponding very irrational numbers $x$ and $p x$ are in one of three symmetric relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Raney Transducers and the Lowest Point of the $p$-Lagrange spectrum Dong, Brandon Dupont, Soren O'Dorney, Evan M. Waitkus, W. Theo Number Theory It is well known that the golden ratio $ϕ$ is the ''most irrational'' number in the sense that its best rational approximations $s/t$ have error $\sim 1/(\sqrt{5} t^2)$ and this constant $\sqrt{5}$ is as low as possible. Given a prime $p$, how can we characterize the reals $x$ such that $x$ and $p x$ are both ''very irrational''? This is tantamount to finding the lowest point of the $p$-Lagrange spectrum $\mathcal{L}_p$ as previously defined by the third author. We describe an algorithm using Raney transducers that computes $\min \mathcal{L}_p$ if it terminates, which we conjecture it always does. We verify that $\min \mathcal{L}_p$ is the square root of a rational number for primes $p < 2000$. Mysteriously, the highest values of $\min \mathcal{L}_p$ occur for the Heegner primes $67$, $3$, and $163$, and for all $p$, the continued fractions of the corresponding very irrational numbers $x$ and $p x$ are in one of three symmetric relations. |
| title | Raney Transducers and the Lowest Point of the $p$-Lagrange spectrum |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.15480 |