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

Zhuowen Guo, Kangbo Ouyang, Jiahao Qiu, Shuhao Zhang

TL;DR

This work extends fractal transference principles to multidimensional digit-restricted continued-fraction sets. By introducing the singular-value potential $\phi^s$ and the multivariate Dirichlet series $\ζ_S(\boldsymbol\sigma)$, the authors obtain universal Hausdorff-dimension bounds for digit-restricted fractals in $[0,1]^d$ and establish sharp results under positive density and uniform $K$-balanced conditions, yielding $\dim_H(\mathcal E_S)=d/2$ in dense cases and $\dim_H(\mathcal E_S^{\mathrm{vec}})=d/(2\alpha)$ in relative settings with polynomial density exponent $\alpha$. The paper also develops a robust higher-dimensional Moran framework and a three-stage seed-insertion-elimination construction to realize configurations inside fractal digit sets without dimension loss, enabling persistence of translation-invariant patterns such as multidimensional Szemerédi-type configurations. Collectively, these results bridge additive-density phenomena in $\mathbb N^d$ with geometric properties of coordinatewise continued-fraction fractals, offering tools and constructions applicable to Diophantine approximation, fractal geometry, and combinatorial number theory.

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 $φ^s(\mathbf a)$ and the multivariate Dirichlet series $ζ_S(\boldsymbolσ) =\sum_{\mathbf a\in S}\prod_{j=1}^d a_j^{-σ_j}$. Let $s_\ast:=\inf\{s>0:\sum_{\mathbf a\in S}φ^s(\mathbf a)<\infty\}$ and $Λ_S:=\inf\{σ_1+\cdots+σ_d:ζ_S(\boldsymbolσ)<\infty\}$. 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.\ $a_n(x_j)\to\infty$ for each $j$), and $s_\ast=\tfrac12Λ_S$ for uniformly $K$--balanced $S$. In particular, if $S\subset\mathbb{N}^d$ has positive upper (or upper Banach) density then $\dim_H(\mathcal E_S)=d/2$. 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.

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

TL;DR

This work extends fractal transference principles to multidimensional digit-restricted continued-fraction sets. By introducing the singular-value potential and the multivariate Dirichlet series , the authors obtain universal Hausdorff-dimension bounds for digit-restricted fractals in and establish sharp results under positive density and uniform -balanced conditions, yielding in dense cases and in relative settings with polynomial density exponent . The paper also develops a robust higher-dimensional Moran framework and a three-stage seed-insertion-elimination construction to realize configurations inside fractal digit sets without dimension loss, enabling persistence of translation-invariant patterns such as multidimensional Szemerédi-type configurations. Collectively, these results bridge additive-density phenomena in with geometric properties of coordinatewise continued-fraction fractals, offering tools and constructions applicable to Diophantine approximation, fractal geometry, and combinatorial number theory.

Abstract

We establish a multidimensional fractal transference principle for digit-restricted sets associated with subsets of , extending the one-dimensional framework of Nakajima--Takahasi, Adv. Math. (2025). We develop general Hausdorff-dimension tools via the singular value potential and the multivariate Dirichlet series . Let and . We obtain , where denotes the set of points whose continued-fraction digit vectors lie in and whose coordinates escape (i.e.\ for each ), and for uniformly --balanced . In particular, if has positive upper (or upper Banach) density then . 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.
Paper Structure (33 sections, 43 theorems, 371 equations)

This paper contains 33 sections, 43 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Coordinatewise continued fractions and digit sets
  3. Fractal transference principles on Nd
  4. Relative fractal transference on Nd
  5. Hausdorff dimension of digit-restricted sets
  6. Organization of the paper
  7. Preliminaries
  8. Some facts on continued fractions
  9. Scaling and reduction lemmas in $\mathbb{N}^d$
  10. Almost Lipschitz maps and elimination maps
  11. Upper bounds on $\dim_H$ for digit--restricted sets in $\mathbb{N}^d$
  12. Main upper bounds
  13. Universal upper bounds via singular values and Dirichlet series
  14. Proof of Theorem \ref{['thm:zeta-vs-singular']}
  15. Proof of Theorem \ref{['thm:full-restriction']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $S\subset\mathbb{N}^d$.

Theorems & Definitions (98)

  • Theorem 1.1: Fractal transference principle on $\mathbb{N}^d$
  • Remark 1.2
  • Corollary 1.3: Polynomial configurations in digit sets
  • Theorem 1.4: Relative fractal transference on $\mathbb{N}^d$
  • Remark 1.5
  • Theorem 1.6: Universal upper bound and sharpness
  • Corollary 1.7: Polynomial growth in the balanced regime
  • Corollary 1.8: Vector-injective subregime
  • Lemma 2.1
  • proof
  • ...and 88 more