Hausdorff measure of sets of inhomogeneous Dirichlet non-improvable affine forms with weights

TL;DR

This work addresses the size of the inhomogeneous Dirichlet non-improvable sets with weights, $D_{{\bm\alpha},{\bm\beta}}^{\mathbf{b}}(\psi)$, in higher dimensions by proving a zero-full law for the corresponding Hausdorff measure under a decay condition on the approximating function $\psi$. The authors leverage a Diophantine transference principle to recast the problem as a limsup set defined by neighborhoods of hyperplanes and then apply the mass transference principle from balls to rectangles to transfer Lebesgue-measure results to Hausdorff $f$-measures. A divergence-part construction yields a full Lebesgue-measure limsup set $W_{n,m}(\Phi)$ that is subsequently converted to a full Hausdorff $f$-measure result via the transference principle, while the convergence part uses a controlled hyperrectangle cover to prove zero $\mathcal{H}^f$-measure. The results generalize weighted inhomogeneous Dirichlet non-improvability and answer a question of Kim and Kim (Adv. Math. 2022), demonstrating a robust zero-full law in this weighted setting.

Abstract

Under a reasonable decay assumption on the approximating function, we establish a zero-full law for the Hausdorff measure of sets of inhomogeneous Dirichlet non-improvable affine forms with weights, thereby answering a question posed by Kim and Kim (§5.3, Adv. Math., 2022).

Theorems & Definitions (32)

• Theorem 1.1: Dirichlet
• Corollary 1.2
• Theorem 1.3: Khintchine--Groshev Theorem
• Remark 1
• Theorem 1.4: Jarnik31DV97
• Theorem 1.5: Kim22
• Theorem 1.6
• Remark 2
• Remark 3
• Theorem 2.1: KR21 and Zh21