Vol. 8, No. 1, 2019

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Admissible endpoints of gaps in the Lagrange spectrum

Dmitry Gayfulin

Vol. 8 (2019), No. 1, 47–56
DOI: 10.2140/moscow.2019.8.47

For any irrational number α define the Lagrange constant μ(α) by

μ1(α) = liminf p,q|q(qα p)|.

The set of all values taken by μ(α) as α varies is called the Lagrange spectrum L. An irrational α is called attainable if the inequality

|α p q| 1 μ(α)q2

holds for infinitely many integers p and q. We call a real number λ L admissible if there exists an irrational attainable α such that μ(α) = λ. In a previous paper we constructed an example of a nonadmissible element in the Lagrange spectrum. In the present paper we give a necessary and sufficient condition for admissibility of a Lagrange spectrum element. We also give an example of an infinite sequence of left endpoints of gaps in L which are not admissible.

Lagrange spectrum, Diophantine approximation, continued fractions
Mathematical Subject Classification 2010
Primary: 11J06
Received: 10 January 2018
Accepted: 17 March 2018
Published: 11 August 2018
Dmitry Gayfulin
Steklov Mathematical Institute
Russian Academy of Sciences