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

Abstract

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 $\zeta_S(\boldsymbol{\sigma}) =\sum_{\mathbf a\in S}\prod_{j=1}^d a_j^{-\sigma_j}$. Let $s_\ast:=\inf\{0<s\le d:\sum_{\mathbf a\in S}\phi^s(\mathbf a)<\infty\}$ and $\Lambda_S:=\inf\{\sigma_1+\cdots+\sigma_d:\zeta_S(\boldsymbol{\sigma})<\infty\}$. We obtain $\dim_H(\mathcal E_S)\le s_\ast$, where $\mathcal E_S\subset((0,1)\setminus\mathbb{Q})^d$ denotes the set of points whose continued-fraction digit vectors lie in $S$ and whose coordinates escape (i.e.\ $a_n(x_j)\to\infty$ for each $j$), and $s_\ast=\tfrac12\Lambda_S$ for uniformly $K$--balanced $S$. If $S\subset\mathbb{N}^d$ has positive upper density, the transference theorem constructs a set $E_S\subset\mathcal E_{\mathbb N^d}^{\mathrm{vec}}$ with $\dim_H E_S=d/2$; in the positive upper Banach density case we can construct $F_S\subset\mathcal E_{\mathbb N^d}^{\mathrm{vec}}$ with $\dim_H F_S=d/2$. In both cases the common digit set recovers the corresponding density of $S$. On the combinatorial side, the transference principle ensures that translation-invariant configurations forced at positive density, including multidimensional Szemer\'edi patterns, persist inside the induced fractal digit sets.