One-sided Diophantine approximations

Abstract

The paper deals with best one-sided (lower or upper) Diophantine approximations of the -th kind ( ∈ N). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction p q ∈ Q to be a best lower or upper Diophantine approximation of the -th kind to a given α ∈ 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.

On Helly numbers of exponential lattices

“…Similarly, p q is a best upper approximation of r if q r − p q < q r − p q ≤ 0 for all rational numbers p q with p q ≥ r, p q = p q , and 0 < q ≤ q. The first part of the following lemma is a classical result, the second and third parts are recent results of Hančl and Turek [8].…”

On Helly numbers of exponential lattices

“…It is a well known fact that convergents are best approximations of r [14]. The following lemma about best lower and upper best approximations is a recent result of Hančl and Turek [10].…”