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In the problem of (simultaneous) Diophantine approximation in $ℝ^{3}$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $K_{2} : (y^{2} + z^{2}) (x^{2} + y^{2} + z^{2}) \leq 1$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $Δ (K_{c})$ of more general star bodies $K_{c} : {(y^{2} + z^{2})}^{c / 2} (x^{2} + y^{2} + z^{2}) \leq 1,$ where $c$ is any positive constant. These are obtained by inscribing into $K_{c}$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$ .

Lower bounds for simultaneous Diophantine approximation constants

Werner Georg Nowak (2014)

Communications in Mathematics

After a brief exposition of the state-of-art of research on the (Euclidean) simultaneous Diophantine approximation constants $θ_{s}$ , new lower bounds are deduced for $θ_{6}$ and $θ_{7}$ .

On a measure theoretic aspect of diophantine approximation.

Stefan Kühnlein (1996)

Journal für die reine und angewandte Mathematik

On ternary cubic forms.

Nowak, Werner Georg (2000)

Mathematica Pannonica

On the critical determinants of certain star bodies

Werner Georg Nowak (2017)

Communications in Mathematics

In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body $| x_{1} | ({| x_{1} |}^{3} + {| x_{2} |}^{3} + {| x_{3} |}^{3}) \leq 1 .$ In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved...

О критическом определителe области ${| x |}^{p} + {| y |}^{p} l t; 1$ . (Дополнения к статье "О применении ЭВМ к доказательству одной гипотезы Минковского из геометрии чисел". Зап. семинаров ЛОМИ, т.71,1977, с.163-180)

А.В. Малышев (1979)

Zapiski naucnych seminarov Leningradskogo

О применении ЭВМ к доказательству одной гипотезы Минковского из геометрии чисел

А.В. Малышев (1977)

Zapiski naucnych seminarov Leningradskogo

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A non-convex generalization of the circle problem.

A Note on Siegel's Lemma Over Number Fields.

A Remark to the Paper of H. Hadwiger "Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl".

An adelic Minkowski-Hlawka theorem and an application to Siegel's lemma.

Ein dreidimensionales Gitterpunktproblem.

Eine neue Abschätzung der kritischen Determinante von Sternkörpern.

Estimating the critical determinants of a class of three-dimensional star bodies

Lower bounds for simultaneous Diophantine approximation constants

On a measure theoretic aspect of diophantine approximation.

On ternary cubic forms.

On the critical determinants of certain star bodies

On the Number of Primitive Lattice Points in Plane Domains.

Quaternary quadratic forms and an associated lattice constant

Über das Mordellsche Umkehrproblem für den Minkowskischen Linearformensatz

Über eine Klasseneinteilung der Sternkörper.

Доказательство гипотезы Минковского о критическом определителе области ${| x |}^{p} + {| y |}^{p} l t; 1$

Замечание о звездном множестве

Метод Морделла взаимных решеток в геометрии чисел

О применении ЭВМ к доказательству одной гипотезы Минковского из геометрии чисел

Currently displaying 1 – 20 of 21