## 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

#### 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

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
Secondary: 11J81: Transcendence (general theory)

#### 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

#### References

• F. Amoroso, f-transfinite diameter and number theoretic applications, Ann. Inst. Fourier Grenoble 43 (1993), 1179–1198.
• J. Aguirre and J.C. Peral, The integer Chebyshev constant of Farey intervals, pp. 11–27 in Proceedings of the Primeras Jornadas de Teoría de Números, special issue of Publ. Mat. (2007).
• G. V. Chudnovsky. Number theoretic applications of polynomials with rational coefficients defined by extremality conditions, pp. 61–105 in Arithmetic and Geometry I, edited by M. Artin and J. Tate, Progress in Math. \bf35, Birkhäuser (1983).
• V. Flammang, Sur le diamètre transfini entier d'un intervalle à extrémités rationnelles, Ann. Inst. Fourier, Grenoble 45 (1995), 779–793.
• V.Flammang, On the absolute length of polynomials having all zeros in a sector, J. Number Theory 143 (2014), 385–401.
• D.Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. \bf18 (1894), 155–159.
• L.B.O. Ferguson, Approximation by polynomials with integral coefficients, Math. Surveys \bf17, American Math. Soc. (1980).
• A. Meichsner, The integer Chebyshev problem: computational explorations, Ph.D. thesis, Simon Fraser University (2009).
• I.E. Pritsker, Small polynomials with integer coefficients, J. Anal. Math. 96 (2005), 151–190.
• C.J. Smyth, On the measure of totally real algebraic numbers I, J. Austral. Math. Soc. A \bf30 (1980), 137–149.
• C.J. Smyth, The mean value of totally real algebraic numbers, Math. Comp. 42 (1984), 663–681.
• Q. Wu, On the linear independence measure of logarithms of rational numbers, Math. Comp. 72 (2003), 901–911.