Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, p. 591-603

In a series of recent papers, W. M. Schmidt and L. Summerer developed a new theory by which they recover all major generic inequalities relating exponents of Diophantine approximation to a point in n , and find new ones. Given a point in n , they first show how most of its exponents of Diophantine approximation can be computed in terms of the successive minima of a parametric family of convex bodies attached to that point. Then they prove that these successive minima can in turn be approximated by a certain class of functions which they call (n,γ)-systems. In this way, they bring the whole problem to the study of these functions. To complete the theory, one would like to know if, conversely, given an (n,γ)-system, there exists a point in n whose associated family of convex bodies has successive minima which approximate that function. In the present paper, we show that this is true for a class of functions which they call regular systems.

Récemment, W. M. Schmidt et L. Summerer ont introduit une nouvelle théorie qui leur permet de redémontrer les principales inégalités connues liant les exposants d’approximation diophantienne d’un point de n , et d’en trouver de nouvelles. Ils montrent d’abord comment la plupart de ces exposants peuvent être calculés en termes des minima successifs d’une famille de convexes à un paramètre attachée à ce point. Puis ils démontrent que ces minima peuvent, à leur tour, être approchés par une certaine classe de fonctions dites de type (n,γ). Ils ramènent ainsi le problème initial à l’étude de ces fonctions. Pour compléter la théorie, on voudrait savoir si, en retour, étant donné une fonction de ce type, il existe un point de n dont les minima de la famille de convexes correspondante approchent cette fonction. On montre ici que tel est le cas pour les fonctions dites régulières.

DOI : https://doi.org/10.5802/jtnb.915
Classification:  11J13,  11J82
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     author = {Roy, Damien},
     title = {Construction of points realizing  the regular systems of  Wolfgang Schmidt and Leonard Summerer},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     pages = {591-603},
     doi = {10.5802/jtnb.915},
     mrnumber = {3393168},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2015__27_2_591_0}
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Roy, Damien. Construction of points realizing  the regular systems of  Wolfgang Schmidt and Leonard Summerer. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 591-603. doi : 10.5802/jtnb.915. http://www.numdam.org/item/JTNB_2015__27_2_591_0/

[1] Y. Bugeaud and M. Laurent, On transfer inequalities in Diophantine approximation II, Math. Z. 265 (2010), 249–262. | MR 2609309 | Zbl 1234.11086

[2] O. N. German, Intermediate Diophantine exponents and parametric geometry of numbers, Acta Arith. 154 (2012), 79–101. | MR 2943676

[3] P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, North-Holland, (1987). | MR 893813 | Zbl 0611.10017

[4] V. Jarník, Zum Khintchineschen Übertragungssatz, Trav. Inst. Math. Tbilissi 3 (1938), 193–212. | Zbl 0019.10602

[5] A. Y. Khintchine, Zur metrischen Theorie der Diophantischen Approximationen, Math. Z. 24 (1926), 706–714. | MR 1544787

[6] A. Y. Khintchine, Über eine Klasse linearer Diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170–195.

[7] M. Laurent, Exponents of Diophantine approximation in dimension two, Can. J. Math. 61 (2009), 165–189. | MR 2488454 | Zbl 1229.11101

[8] N. G. Moshchevitin, Exponents for three-dimensional simultaneous Diophantine approximations, Czechoslovak Math. J. 62 (2012), 127–137. | MR 2899740 | Zbl 1249.11061

[9] D. Roy, On Schmidt and Summerer parametric geometry of numbers, Ann. of Math. 182 (2015), to appear.

[10] W. M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Ann. of Math. 85 (1967), 430–472. | MR 213301 | Zbl 0152.03602

[11] W. M. Schmidt and L. Summerer, Parametric geometry of numbers and applications, Acta Arith. 140 (2009), 67–91. | MR 2557854 | Zbl 1236.11060

[12] W. M. Schmidt and L. Summerer, Diophantine approximation and parametric geometry of numbers, Monatsh. Math. 169 (2013), 51–104. | MR 3016519 | Zbl 1264.11056

[13] W. M. Schmidt and L. Summerer, Simultaneous approximation to three numbers, Mosc. J. Comb. Number Theory 3 (2013), 84–107. | MR 3284111 | Zbl 1301.11058