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2019 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
Rocky Mountain J. Math. 49(5): 1547-1562 (2019). DOI: 10.1216/RMJ-2019-49-5-1547

## 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 $|as-bq|=1$.

## Citation

V. Flammang. "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 (5) 1547 - 1562, 2019. https://doi.org/10.1216/RMJ-2019-49-5-1547

## Information

Published: 2019
First available in Project Euclid: 19 September 2019

zbMATH: 07113698
MathSciNet: MR4010572
Digital Object Identifier: 10.1216/RMJ-2019-49-5-1547

Subjects:
Primary: 11K50
Secondary: 11J81  