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Abstract
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We study the property of uniform discreteness within discrete orbits of nonuniform lattices
in
, acting
on
by
linear transformations. We provide quantitative consequences of previous results by
using Diophantine properties. We give a partial result toward a conjecture of Lelièvre
regarding the set of long cylinder holonomy vectors of the “golden L” translation surface:
for any
,
three points of this set can be found on a horizontal line within a distance of
of
each other.
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Keywords
golden L, discrete orbit, holonomy vectors, nonuniform
lattices, convergents of the continued fraction, Samuel
Lelièvre conjecture
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Mathematical Subject Classification
Primary: 37B05
Secondary: 06B25, 11J70
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Milestones
Received: 11 September 2024
Revised: 18 September 2025
Accepted: 4 October 2025
Published: 18 November 2025
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| © 2025 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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