Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

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Abstract

Let M be an m-dimensional, Ck manifold in Rn, for any k, m, n ? ?, and for any t> 0 let Lt(M)={x?M : ?qx?<q-t for infinitely many q??}, where, for x?R, ?x? min{ x - i :i?Z}, and for x=(x1,?,xn)???n, ?X? = max{?x1pr,?,?xn?}. In this paper it will be shown that for any Ck manifold M there exist Ck manifolds Mz, Mp arbitrarily 'Ck-close' to M with the property that, for all sufficiently large t, dimLt(Mz)=0, dim Lt(Mp)>0. This result shows that the non-zero curvature conditions which have been successfully used to tackle other aspects of the theory of Diophantine approximation on manifolds are unable to distinguish between these two cases when we look at simultaneous Diophantine approximation. © 2002 Elsevier Science (USA). All rights reserved.

Original languageEnglish
Pages (from-to)1-9
Number of pages9
JournalJournal of Number Theory
Volume98
Issue number1
DOIs
Publication statusPublished - 1 Jan 2003

Keywords

  • Hausdorff dimension
  • Manifolds
  • Simultaneous Diophantine approximation

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