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arXiv · 1809.01013

One-sided Diophantine approximations

Jaroslav Hančl·Ondřej Turek
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Abstract

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 $α\in\mathbb{R}$. 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.

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BibTeXRIS

Jaroslav Hančl, Ondřej Turek. 2019-01-15. One-sided Diophantine approximations. https://doi.org/10.1088/1751-8121%2Faaf5d3

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