One-sided Diophantine approximations
| Autor: | Ondřej Turek, Jaroslav Hančl |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Approximations of π General Physics and Astronomy FOS: Physical sciences Diophantine approximation 01 natural sciences Combinatorics Mathematics - Spectral Theory FOS: Mathematics Fraction (mathematics) Spectral analysis Number Theory (math.NT) 0101 mathematics Spectral Theory (math.SP) Mathematical Physics Mathematics 11J70 (Primary) 81Q35 11K60 (Secondary) Mathematics - Number Theory Diophantine equation 010102 general mathematics Statistical and Nonlinear Physics Mathematical Physics (math-ph) 010101 applied mathematics One sided Modeling and Simulation Quantum graph Metric (mathematics) |
| DOI: | 10.48550/arxiv.1809.01013 |
| Popis: | The paper deals with best one--sided (lower or upper) Diophantine approximations of the $\ell$-th kind ($\ell\in\mathbb{N}$). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction $\frac{p}{q}\in\mathbb{Q}$ to be a best lower or upper Diophantine approximation of the $\ell$-th kind to a given $\alpha\in\mathbb{R}$. The sets of best lower and upper approximations are examined in terms of their cardinalities and metric properties. Applying our results in spectral analysis, we obtain an explanation for the rarity of so-called Bethe--Sommerfeld quantum graphs. Comment: 24 pages, 2 figures; revised version |
| Databáze: | OpenAIRE |
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