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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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

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

First available in Project Euclid: 19 September 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

integer transfinite diameter auxiliary functions recursive algorithm


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.

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