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2019 Admissible endpoints of gaps in the Lagrange spectrum
Dmitry Gayfulin
Mosc. J. Comb. Number Theory 8(1): 47-56 (2019). DOI: 10.2140/moscow.2019.8.47

Abstract

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

μ 1 ( α ) = liminf p , q | q ( q α 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 ) q 2

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.

Citation

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Dmitry Gayfulin. "Admissible endpoints of gaps in the Lagrange spectrum." Mosc. J. Comb. Number Theory 8 (1) 47 - 56, 2019. https://doi.org/10.2140/moscow.2019.8.47

Information

Received: 10 January 2018; Accepted: 17 March 2018; Published: 2019
First available in Project Euclid: 3 December 2018

zbMATH: 07063262
MathSciNet: MR3864307
Digital Object Identifier: 10.2140/moscow.2019.8.47

Subjects:
Primary: 11J06

Rights: Copyright © 2019 Mathematical Sciences Publishers

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