A problem in non-linear Diophantine approximation. (27th March 2018)

• Record Type: Journal Article
• Authors: Harrap, Stephen; Hussain, Mumtaz; Kristensen, Simon
• Is Part Of: Nonlinearity. Volume 31:Number 5(2018:May)
• Page Start: 1734
• Page End: 1756
• Publication Date: 2018-03-27
• Subjects: non-linear Diophantine approximation -- partial differential equations -- Hausdorff measure -- Hausdorff dimension -- Khintchine–Groshev theorem; 11K60 -- 11J83 -- 11J86 -- 37A45; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aaa498
• Languages: English
• ISSNs: 0951-7715

Abstract

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.