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 ${\varphi }^s(\mathbf{a})$ and the multivariate Dirichlet series ${\zeta }_S(\mathbf{\sigma })={\sum }_{\mathbf{a}\in S}{\prod }_{j=1}^d{a}_j^{-{\sigma }_j}$. Let $s_{\ast }:=\inf \{0<s\le d:{\sum }_{\mathbf{a}\in S}{\varphi }^s(\mathbf{a})<\infty \}$ and ${\Lambda }_S:=\inf \{{\sigma }_1+\cdots +{\sigma }_d:{\zeta }_S(\mathbf{\sigma })<\infty \}$. 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 $s_{\ast }=\frac{1}{2}{\Lambda }_S$ 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 Szemerédi patterns, persist inside the induced fractal digit sets.

v3 (50 pages): Revised the Hausdorff-dimension statements for digit-restricted sets and corrected several technical details in the proofs. The main fractal transference theorems are unchanged