Fractal transference principles for subsets of ${\mathbb{N}}^d$ of positive density

We establish a multidimensional fractal transference principle for digit-restricted sets associated with subsets of ${\mathbb{N}}^d$, extending the one-dimensional framework of Nakajima–Takahasi, Adv. Math. (2025). We develop general Hausdorff-dimension tools via the singular value potential ${\phi }^s(\mathbf{a})$ and the multivariate Dirichlet series ${\zeta }_S(\mathit{\sigma })={\sum }_{\mathbf{a}\in S}{\prod }_{j=1}^d{a}_j^{-{\sigma }_j}$. Let $s_{\ast }:=\inf \{s>0:{\sum }_{\mathbf{a}\in S}{\phi }^s(\mathbf{a})<\mathrm{\infty }\}$ and ${\mathrm{\Lambda }}_S:=\inf \{{\sigma }_1+\cdots +{\sigma }_d:{\zeta }_S(\mathit{\sigma })<\mathrm{\infty }\}$. We obtain ${\dim }_H({\mathcal{E}}_S)\le s_{\ast }$, where ${\mathcal{E}}_S\subset (0,1{)}^d$ denotes the set of points whose continued-fraction digit vectors lie in $S$ and whose coordinates escape (i.e.\ $a_n(x_j)\to \mathrm{\infty }$ for each $j$), and $s_{\ast }=\frac{1}{2}{\mathrm{\Lambda }}_S$ for uniformly $K$–balanced $S$. In particular, if $S\subset {\mathbb{N}}^d$ has positive upper (or upper Banach) density then ${\dim }_H({\mathcal{E}}_S)=d\mathrm{/}2$. On the combinatorial side, the transference principle ensures that translation-invariant configurations forced at positive density, including multidimensional Szemerédi patterns, persist inside the induced fractal digit sets.