Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension

Victor Beresnevich, Lawrence Lee, Robert C. Vaughan, Sanju Velani

Research output: Contribution to journalArticle

Abstract

Given n ϵ ℕ and τ > 1/n, let Snn.(τ) denote the classical set of τ-approximable points in ℝn, which consists of x ϵ ℝn that lie within distance q-τ-1 from the lattice .(1/q)ℤn for infinitely many q ϵ ℕ. In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ℳ of ℝn and any τ > 1/n almost all points on ℳ are not τ-Approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ℳ ∩ Sn(τ). In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971-992], which enables us to find a sharp lower bound for dimℳ Sn(τ) for any C2 submanifold ℳ of ℝn and any τ satisfying 1/n 6 ≥ τ 1/m. Here m is the codimension ofM.We also show that the condition on τ is best possible and extend the result to general approximating functions.

Original languageEnglish (US)
Pages (from-to)762-779
Number of pages18
JournalMathematika
Volume63
Issue number3
DOIs
StatePublished - Jan 1 2017

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Diophantine Approximation
Hausdorff Measure
Hausdorff Dimension
Submanifolds
Lower bound
Null set or empty set
Strengthening
Set of points
Codimension
Denote

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Beresnevich, Victor ; Lee, Lawrence ; Vaughan, Robert C. ; Velani, Sanju. / Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension. In: Mathematika. 2017 ; Vol. 63, No. 3. pp. 762-779.
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Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension. / Beresnevich, Victor; Lee, Lawrence; Vaughan, Robert C.; Velani, Sanju.

In: Mathematika, Vol. 63, No. 3, 01.01.2017, p. 762-779.

Research output: Contribution to journalArticle

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