Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension

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

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3 Scopus citations

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 - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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