Parametric geometry of numbers and applications

Schmidt, Wolfgang M.; Summerer, Leonhard

doi:10.4064/aa140-1-5

Cited by 81 publications

(190 citation statements)

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“…[155,156,157]), where they achieved a considerable progress on old questions of Jarnik using their parametric geometry of numbers, even if these results clearly belong to Diophantine number theory.…”

Section: Who We Are -The Research Groups We Belong Tomentioning

confidence: 99%

30 years of collaboration

Fuchs

Hajdu

2016

Period Math Hung

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Abstract. We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn.

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Section: Who We Are -The Research Groups We Belong Tomentioning

confidence: 99%

30 years of collaboration

Fuchs

Hajdu

2016

Period Math Hung

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“…(Более общее утверждение и приложения к затронутым в настоящей статье вопросам, в частности, следствия для теории переноса, имеются в недавней работе В. М. Шмидта и Л. Саммерера [64].) В частности, при n = 1 из теоремы Минковского о последовательных минимумах получаем, что…”

Section: задача о последовательных минимумах пусть даны решеткаunclassified

“…-Теоремы переноса для сингулярных систем (работы В. Ярника [9], А. Ап-фельбека [29], М. Лорана и Я. Бюжо [43], [68], [69]; здесь же уместно упомянуть недавнюю работу В. М. Шмидта и Л. Саммерера [64]). …”

Section: теоремы переносаunclassified

Сингулярные Диофантовы Системы А. Я. Хинчина И Их Применение

Moshchevitin¹

2010

Uspekhi Mat. Nauk

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“…M. Schmidt and L. Summerer [3,4] studied the asymptotic behaviour of the successive minima of the body G t B d ∞ with respect to the given lattice Λ. An appropriate choice of Λ connects this setting with the classical setting of simultaneous Diophantine approximation.…”

Section: Introductionmentioning

confidence: 99%

A simple proof of Schmidt–Summerer’s inequality

German

Moshchevitin

2012

Monatsh Math

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In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W. M. Schmidt and L. Summerer. Introduction∞ the unit ball in sup-norm, i.e. the cube with vertices at the points (±1, . . . , ±1).W. M. Schmidt and L. Summerer [3,4] studied the asymptotic behaviour of the successive minima of the body G t B d ∞ with respect to the given lattice Λ. An appropriate choice of Λ connects this setting with the classical setting of simultaneous Diophantine approximation.In [4] Schmidt and Summerer proved important inequalities connecting the asymptotics of the first and the p-th successive minima, which lead them to an improvement of a famous Jarník's inequality between the uniform and the ordinary Diophantine exponents [2]. However, the proof they proposed was rather difficult. It uses Mahler's theory of compound bodies and involves a complicated, cumbersome analysis of special piecewise linear functions.In the present paper we give a short proof of the main result of [4]. It relies on a simple geometric observation (see Lemma 1 below) and does not use the theory of compounds.

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Parametric geometry of numbers and applications

Cited by 81 publications

References 5 publications

30 years of collaboration

30 years of collaboration

Сингулярные Диофантовы Системы А. Я. Хинчина И Их Применение

A simple proof of Schmidt–Summerer’s inequality

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