One-sided Diophantine approximations

TL;DR

This paper characterizes best one-sided Diophantine approximations using continued fractions, analyzes their properties, and applies findings to spectral analysis, explaining the rarity of certain quantum graphs.

Contribution

It provides explicit criteria for best one-sided approximations of the $ ext{l}$-th kind and explores their properties, linking number theory with spectral analysis.

Findings

Explicit criteria for best approximations using continued fractions

Analysis of the cardinality and metric properties of approximation sets

Application to explaining the rarity of Bethe–Sommerfeld quantum graphs

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 $\alpha\in\mathbb{R}$. 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.