A problem in non-linear Diophantine approximation

TL;DR

This paper investigates the measure-theoretic properties of vectors satisfying non-linear Diophantine inequalities linked to PDE solvability, revealing measure results based on Diophantine conditions.

Contribution

It provides new Lebesgue and Hausdorff measure results for sets defined by non-linear Diophantine inequalities related to PDEs, connecting Diophantine conditions with solution existence.

Findings

Lebesgue measure results for the set of solutions

Hausdorff measure results for the solution set

Diophantine condition failure implies smooth solutions

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.