Optimality of two inequalities for exponents of Diophantine approximation

Schleischitz, Johannes

doi:10.1016/j.jnt.2022.09.003

Cited by 4 publications

(1 citation statement)

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“…Laurent [42] proved that for n = 2 these inequalities are sharp. However, for n ⩾ 3 it was shown by Schleischitz [43] that inequalities (24) are no longer sharp. This is not surprising in the light of the following inequalities, obtained in 2013 by Schmidt and Summerer [44] (also see [45] and [46], where shorter proofs of their theorem are proposed).…”

Section: 31mentioning

confidence: 99%

Geometry of Diophantine exponents

German

2023

Russian Math. Surveys

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Diophantine exponents are some of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation that studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle. Bibliography: 99 titles.

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Section: 31mentioning

confidence: 99%

Geometry of Diophantine exponents

German

2023

Russian Math. Surveys

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On the transference principle and Nesterenko's linear independence criterion

German¹,

Moshchevitin²

2023

Известия Российской академии наук. Серия математическая

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We consider the problem of simultaneous approximation of real numbers $\theta_1,…,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+…+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion. Bibliography: 19 titles.

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On the transference principle and Nesterenko's linear independence criterion

German

Moshchevitin

2023

Izv. Math.

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We consider the problem of simultaneous approximation of real numbers $\theta_1, …,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+…+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.

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Optimality of two inequalities for exponents of Diophantine approximation

Cited by 4 publications

References 42 publications

Geometry of Diophantine exponents

Geometry of Diophantine exponents

On the transference principle and Nesterenko's linear independence criterion

On the transference principle and Nesterenko's linear independence criterion

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