{\mathbb{Z}}^n \backslash \{0\}} (\Psi({\mathbf{a}}))^{s+1-m}=\infty.$$

Given an approximating function $\psi$, the lower order at infinity $\tau_{\psi}$ of $1/\psi$ is defined by $$\tau_{\psi} := \liminf_{t \to \infty}\frac{-\log\psi(t)}{\log t}.$$ The divergent sum condition of Theorem \[thm:divergence\] is satisfied whenever $$s<m-1+\frac{n+1}{\tau_\psi}.$$ Therefore, by the definition of Hausdorff measure and dimension, we get

[\[jar\]]{} Let ${\mathbf{f}}$ and $\Theta$ be as in Theorem \[thm:divergence\]. Let $\psi$ be an approximating function as in (\[def:newpsi\]) such that $n+1\leq \tau_\psi<\infty$. Then $$\dim (\mathcal{W}^{\mathbf{f}}_{(\Psi,\Theta)}\cap{\mathbf{U}})\geq m-1+\frac{n+1}{\tau_\psi}.$$

Remarks
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1.  We have assumed $S$ contains the infinite place in Theorem \[thm:main\]. This is not a serious assumption, the proof in the case when $S$ contains only finite places needs some minor modifications but follows the same outline, details will appear in [@Datta], the PhD thesis, under preparation, of the first named author. In [@MoS1], the (homogeneous) $S$-adic convergence case is proved in slightly greater generality than in the present paper. Namely, instead of ${\mathbb{Q}}$, the quotient field of a finitely generated subring of ${\mathbb{Q}}$ is considered. This, more general formulation will also be investigated in [@Datta].

2.  Our proof for the convergence case, namely Theorem \[thm:main\] blends techniques from the homogeneous results, namely [@KT; @BKM; @MoS1] and uses the transference principle developed 