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Paper

A problem in non-linear Diophantine approximation

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Published 27 March 2018 • © 2018 IOP Publishing Ltd & London Mathematical Society
, , Citation Stephen Harrap et al 2018 Nonlinearity 31 1734 DOI 10.1088/1361-6544/aaa498

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Author e-mails

s.g.harrap@durham.ac.uk

m.hussain@latrobe.edu.au

sik@math.au.dk

Author affiliations

1 Department of Mathematical Sciences, Science Laboratories, Durham University, South Rd, Durham, DH1 3LE, United Kingdom

2 Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo, Vic 3552, Australia

3 Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

Author notes

4 MH's research is supported by the Australian Research Council and La Trobe University's start-up grant.

5 SK's research is supported by the Danish Natural Science Research Council.

Dates

  1. Received 4 November 2016
  2. Accepted 2 January 2018
  3. Published

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0951-7715/31/5/1734

Abstract

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

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10.1088/1361-6544/aaa498