On Schmidt and Summerer parametric geometry of numbers

• 成果类型: Article
• 署名作者: Roy, Damien
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2015.182.2.9
• 发表日期: 2015
• 页码: 739-786
• 关键词: diophantine approximations; exponents

摘要

Recently, W. M. Schmidt and L. Summerer introduced a new theory which allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in \mathbb{R}boolean AND n, and to discover new ones. They first note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point. Then they prove that the n-tuple of these successive minima can in turn be approximated up to bounded difference by a function from a certain class. In this paper, we show that the same is true within a smaller and simpler class of functions which we call rigid systems. We also show that conversely, given a rigid system, there exists a point in \mathbb{R}boolean AND n whose associated family of convex bodies has successive minima which approximate that rigid system up to bounded difference. As a consequence, the problem of describing the joint spectrum of a family of exponents of Diophantine approximation is reduced to combinatorial analysis.