Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer
Journal de Théorie des Nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 591-603.

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.

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.

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

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