The metrical theory  of simultaneously small linear forms

Hussain, Mumtaz; Levesley, Jeremy

doi:10.7169/facm/2013.48.2.1

Cited by 17 publications

(25 citation statements)

References 10 publications

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“…Then, since |q| ≤ |q| 2 , we have that 1 m ) ∩ I nm | = 1. Further, note that, by (14), Υ p,q → 0 as |q| → ∞. Therefore, Theorem 1 is applicable with k = nm, l = m(n − 1) and m and we conclude that for any ball B ⊂ I nm we have that H f (B ∩ Λ(Υ)) = H f (B).…”

Section: Consider λ(G(υ)mentioning

confidence: 61%

A mass transference principle for systems of linear forms and its applications

Allen¹,

Beresnevich²

2018

Compositio Math.

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In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].

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Section: Consider λ(G(υ)mentioning

confidence: 61%

A mass transference principle for systems of linear forms and its applications

Allen¹,

Beresnevich²

2018

Compositio Math.

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“…These theorems were put in a more general context in [3]; also in that paper some of the conditions on the dimension and approximating functions (used in [11,15,18,19]) were shown to be unnecessary. When u = 0 Theorem 2 reduces to [17,Theorem 1].…”

Section: The Case M + U > Nmentioning

confidence: 99%

The Metric Theory of Mixed Type Linear Forms

Dickinson

Hussain

2012

Int. J. Number Theory

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In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophantine approximation, Acta Arith.68(2) (1994) 133–140] as special cases.

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“…This version is tailored for our use, and is a key ingredient in the proof of Theorem 2. The slicing technique is broad-ranging and has been useful in proving several metrical results; for examples, see Hussain and Kristensen [21,22] and Hussain and Levesley [23].…”

Section: Preliminariesmentioning

confidence: 99%

A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

2016

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Let β > 1 be a real number and define the β-transformation on [0, 1] by T β : x → βx mod 1. Further, definefor infinitely many n} andfor infinitely many n}, where Ψ : N → R >0 is a positive function such that Ψ(n) → 0 as n → ∞. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

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The metrical theory of simultaneously small linear forms

Cited by 17 publications

References 10 publications

A mass transference principle for systems of linear forms and its applications

A mass transference principle for systems of linear forms and its applications

The Metric Theory of Mixed Type Linear Forms

A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

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