Inhomogeneous theory of dual Diophantine approximation on manifolds (Q1929188)

In this well-written and technically challenging paper, the authors prove two main theorems concerning inhomogeneous Diophantine approximation on manifolds. Both problems are concerned with the set of solutions to the Diophantine inequality \[ \| {\mathbf a} \cdot {\mathbf y} + \theta({\mathbf y}) \| < \Psi({\mathbf a}), \] where \({\mathbf x}\) lies on some manifold \({\mathcal M} \subseteq {\mathbb R}^n\), \({\mathbf a} \in {\mathbb Z}^n \setminus \{0\}\), \(\theta\) is a real function on \({\mathbb R}^n\) whose restriction to \({\mathcal M}\) is sufficiently smooth and \(\Psi : {\mathbb Z}^n \rightarrow {\mathbb R}_+\) is an approximating function satisfying certain technical conditions. The problem is related to approximation of a real number by algebraic integers. It is shown that if \({\mathcal M}\) is sufficiently non-degenerate and \(\sum_{{\mathbf a} \in {\mathbb Z^n}\setminus \{0\}} \Psi ({\mathbf a}) < \infty\), the Lebesgue measure of the set points \({\mathbf z} \in {\mathcal M}\) satisfying infinitely many of these inequalities is zero. Conversely, it is shown that if \(\Psi\) satisfies an additional technical condition and \[ \sum_{{\mathbf a} \in {\mathbb Z^n}\setminus \{0\}} | {\mathbf a} | (\Psi ({\mathbf a})/| {\mathbf a})^{s+1-m} = \infty, \] the \(s\)-dimensional Hausdorff measure of this set of points is the same as the \(s\)-dimensional Hausdorff measure of \({\mathcal M}\) whenever \(s > m-1\). When \(s = m\), the \(s\)-dimensional Hausdorff measure is comparable to Lebesgue measure on \({\mathcal M}\), and the second result becomes a converse to the first. The conditions on the non-degeneracy of \({\mathcal M}\) and smoothness of \(\theta\) are stronger in the first result than in the second. On the other hand, the conditions on the approximating function \(\Psi\) are considerably stronger in the second result. The proofs are very technical and rely on many recent developments in the theory of metric Diophantine approximation.