Generalization of a density theorem of Khinchin and diophantine approximation
József Beck1 ; William W. L. Chen2
1 Department of Mathematics Hill Center for the Mathematical Sciences Rutgers University Piscataway NJ 08854, USA
2 School of Mathematical and Physical Sciences Faculty of Science and Engineering Macquarie University Sydney NSW 2109, Australia
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 511-542.

La version continue d’un résultat célèbre de Khinchin dit qu’une demi-droite torique dans le carré unitaire [0,1] 2 est superdense, c’est-à-dire vérifie une forme optimale de densité quantitative en temps, si et seulement si la pente de la géodésique est un nombre mal approché. Nous étendons ce résultat de Khinchin au cas où le tore unitaire [0,1] 2 est remplacé par une surface à petits carreaux finie. En particulier, nous montrons qu’il est possible d’étudier ce problème très théorique en se limitant à des outils traditionnels de la théorie des nombres, en utilisant uniquement les fractions continues et le célèbre théorème des trois distances d’approximation diophantienne combiné avec un processus itératif.

Nous améliorons un résultat antérieur des auteurs et de Yang [1] où il est montré que les nombres mal approchables satisfaisant une restriction technique assez sévère sur les chiffres de leurs fractions continues fournissent des géodésiques superdenses. Nous surmontons ici cet obstacle technique.

Cet article est autosuffisant et n’exige pas la connaissance de la théorie des systèmes dynamiques.

The continuous version of a famous result of Khinchin says that a half-infinite torus line in the unit square [0,1] 2 exhibits superdensity, a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. We extend this result of Khinchin to the case when the unit torus [0,1] 2 is replaced by a finite polysquare translation surface, or square tiled surface. In particular, we show that it is possible to study this very number-theoretic problem by restricting to traditional tools in number theory, using only continued fractions and the famous 3-distance theorem in diophantine approximation combined with an iterative process.

We improve on an earlier result of the authors and Yang [1] where it is shown that badly approximable numbers that satisfy a quite severe technical restriction on the digits of their continued fractions lead to superdense geodesics. Here we overcome this technical impediment.

This paper is self-contained, and the reader does not need any knowledge of dynamical systems.

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DOI : 10.5802/jtnb.1255
Classification : 11K38, 37E35
Mots clés : geodesics, billiards, density
József Beck 1 ; William W. L. Chen 2

1 Department of Mathematics Hill Center for the Mathematical Sciences Rutgers University Piscataway NJ 08854, USA
2 School of Mathematical and Physical Sciences Faculty of Science and Engineering Macquarie University Sydney NSW 2109, Australia
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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József Beck; William W. L. Chen. Generalization of a density theorem of Khinchin and diophantine approximation. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 511-542. doi : 10.5802/jtnb.1255. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1255/

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