DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION. Issue 3 (29th November 2017)

• Record Type: Journal Article
• Authors: Beresnevich, Victor; Lee, Lawrence; Vaughan, Robert C.; Velani, Sanju
• Is Part Of: Mathematika. Volume 63:Issue 3(2017)
• Page Start: 762
• Page End: 779
• Publication Date: 2017-11-29
• Subjects: 11JXX (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579317000171
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract : Given n ∈ N and τ > 1 / n, let S n ( τ ) denote the classical set of τ ‐approximable points in R n, which consists of x ∈ R n that lie within distance q − τ − 1 from the lattice ( 1 / q ) Z n for infinitely many q ∈ N . In pioneering work, Kleinbock and Margulis showed that for any non‐degenerate submanifold M of R n 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 ∩ S n ( 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 dim M ∩ S n ( τ ) for any C 2 submanifold M of R n and any τ satisfying 1 / n ⩽ τ < 1 / m . Here m is the codimension of M . We also show that the condition on τ is best possible and extend the result to general approximating functions.