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Abstract
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For any irrational number
define the Lagrange constant
by
The set of all values taken by
as
varies is called the
Lagrange spectrum .
An irrational
is called attainable if the inequality
holds for infinitely many integers
and
. We call a
real number
admissible if there exists an irrational attainable
such
that
.
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
which
are not admissible.
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Keywords
Lagrange spectrum, Diophantine approximation, continued
fractions
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Mathematical Subject Classification 2010
Primary: 11J06
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Milestones
Received: 10 January 2018
Accepted: 17 March 2018
Published: 11 August 2018
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