Rocky Mountain Journal of Mathematics

On the integer transfinite diameter of intervals of the form $ \left [ \frac {r}{s}, u \right ] $ or $[0, (\sqrt {a}- \sqrt {b})^2 ]$ and of Farey intervals

V. Flammang

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Abstract

We consider intervals of the form $\left [ \frac {r}{s}, u \right ] $, where $r,s$ are positive integers with gcd($r,s$)=1 and $u$ is a real number, or of the form $[0, \smash {(\sqrt {a}{-}\sqrt {b})^2} ]$, where $a,b$ are positive integers. Thanks to a lemma of Chudnovsky, we give first a lower bound of the integer transfinite diameter of such intervals. Then, using the method of explicit auxiliary functions and our recursive algorithm, we explain how to get an upper bound for this quantity. We finish with some numerical examples. Secondly, we prove inequalities on the integer transfinite diameter of Farey intervals, i.e., intervals of the type $\bigl [ \frac {a}{q}, \frac {b}{s} \bigr ] $, where $|as-bq|=1$.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1547-1562.

Dates
First available in Project Euclid: 19 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1568880090

Digital Object Identifier
doi:10.1216/RMJ-2019-49-5-1547

Mathematical Reviews number (MathSciNet)
MR4010572

Zentralblatt MATH identifier
07113698

Subjects
Primary: 11K50: Metric theory of continued fractions [See also 11A55, 11J70]
Secondary: 11J81: Transcendence (general theory)

Keywords
integer transfinite diameter auxiliary functions recursive algorithm

Citation

Flammang, V. On the integer transfinite diameter of intervals of the form $ \left [ \frac {r}{s}, u \right ] $ or $[0, (\sqrt {a}- \sqrt {b})^2 ]$ and of Farey intervals. Rocky Mountain J. Math. 49 (2019), no. 5, 1547--1562. doi:10.1216/RMJ-2019-49-5-1547. https://projecteuclid.org/euclid.rmjm/1568880090


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