2005/03/04 by Victor Beresnevich, Beresnevich, Victor, Sanju Velani +1 · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.math/0503078
23 pages
arxiv created 2005/03/04 · arxiv updated 2009/12/01
Let \cSn(ψ1,...,ψn) denote the set of simultaneously (ψ1,...,ψn)--approximable points in \Rn and \cSMn(ψ) denote the set of multiplicatively ψ--approximable points in \Rn. Let \cM be a manifold in \Rn. The aim is to develop a metric theory for the sets \cM ∩ \cSn(ψ1,...,ψn) and \cM ∩ \cSMn(ψ) analogous to the classical theory in which \cM is simply \Rn. In this note, we mainly restrict our attention to the case that \cM is a planar curve \cC. A complete Hausdorff dimension theory is established for the sets \cC ∩ \cS2(ψ1,ψ2) and \cC ∩ \cSM2(ψ) . A divergent Khintchine type result is obtained for \cC ∩ \cS2(ψ1,ψ2) ; i.e. if a certain sum diverges then the one--dimensional Lebesgue measure on \cC of \cC ∩ \cS2(ψ1,ψ2) is full. Furthermore, in the case that \cC is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for \cC ∩ \cS2(ψ1,ψ2) naturally generalize the dimension and Lebesgue measure statements of \citeBDV03. Within the multiplicative framework, our results for \cC ∩ \cSM2(ψ) constitute the first of their type.