On Diophantine transference principles

Ghosh, Anish; Marnat, Antoine

doi:10.1017/s0305004118000014

Cited by 9 publications

(11 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For real vectors and , this provides a lower bound for the uniform simultaneous inhomogeneous exponent in terms of the dual exponent . There have since been refinements and generalisations by a number of authors, among them Beresnevich and Velani [BV10], Ghosh and Marnat [GM19], and Chow et al [CGGMS].…”

Section: Diophantine Exponents and Transference Inequalitiesmentioning

confidence: 99%

Higher-rank Bohr sets and multiplicative diophantine approximation

Chow

Technau

2019

Compositio Math.

View full text Add to dashboard Cite

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

show abstract

Section: Diophantine Exponents and Transference Inequalitiesmentioning

confidence: 99%

Higher-rank Bohr sets and multiplicative diophantine approximation

Chow

Technau

2019

Compositio Math.

View full text Add to dashboard Cite

show abstract

“…Corollaries 1 and 2 combined with (14) provide the following two results. A similar approach was used in [1,9] in the "nonweighted" case.…”

Section: Case Of One Linear Form and Marnat's Examplesmentioning

confidence: 99%

Transference Theorems for Diophantine Approximation With Weights

German

2020

Mathematika

View full text Add to dashboard Cite

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalise the corresponding inequalities that exist in the "nonweighted" case. §1. Introduction. In 1926 Khintchine in his seminal paper [11] proved the famous transference inequalities connecting two dual problems. The first one concerns simultaneous approximation of given real numbers θ 1 , . . . , θ n by rationals, the second one concerns approximating zero with the values of the linear form θ 1 x 1 + · · · + θ n x n + x n+1 at integer points. Later on, Khintchine's inequalities were generalised to the case of several linear forms by Dyson [5]. Given a matrix

show abstract

Section: Remarksmentioning

confidence: 99%

“…It is worth mentioned that in [19], it has been shown that the homogeneous to inhomogeneous transference principle of [8] is flexible enough to be used for arbitrary Diophantine exponents, not just the critical or "extremal" one. As demonstrated in [19], this naturally extends the scope of potential applications of the original transference principle. Beyond extremal statements such as (1.7), the complete inhomogeneous version of the Khintchine-Groshev theorem, both convergence and divergence cases, for nondegenerate manifolds is established in [1].…”

Section: Remarksmentioning

confidence: 99%