Density combinatorics theorems in fractal dimension theory of continued fractions

TL;DR

The paper develops a fractal transference framework that converts density-theoretic statements about subsets of $\mathbb{N}$ into sharp fractal-dimension statements about sets of irrationals via continued fraction digits. It constructs a three-step Moran-fractal procedure to produce $E_{S,A}$ with prescribed Hausdorff dimension $1/(2\alpha)$ (or $1/2$ in the basic case) while preserving density information of the digits, and it demonstrates that arithmetic and polynomial progressions in $S$ induce corresponding patterns among the partial quotients of numbers in a large-dimensional subset of $E$. The results apply to primes (including primes of quadratic form), Piatetski–Shapiro sequences, and other structured sets, and they extend to general $d$-decaying IFSs and semi-regular continued fractions. Overall, the work bridges density combinatorics and fractal geometry of continued fractions, yielding new 1/2-dimensional and relative-density phenomena for digit sequences with broad combinatorial consequences.

Abstract

We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of subsets of irrationals in $(0,1)$ for which the set $\{a_n(x)\colon n\in\mathbb N\}$ of partial quotients induces an injection $n\in\mathbb N\mapsto a_n(x)\in\mathbb N$. Let $(*)$ be a certain property that holds for any subset of $\mathbb N$ with positive upper density. The principle asserts that for any subset $S$ of $\mathbb N$ with positive upper density, there exists a set $E_S$ of Hausdorff dimension $1/2$ such that the set $\bigcup_{n\in\mathbb N}\bigcap_{x\in E_S}\{a_n(x)\}\cap S$ has the same upper density as that of $S$, and thus inherits property $(*)$. Examples of $(*)$ include the existence of arithmetic progressions of arbitrary lengths and the existence of arbitrary polynomial progressions, known as Szemerédi's and Bergelson-Leibman's theorems respectively. In the same spirit, we establish a relativized version of the principle applicable to the primes, to the primes of the form $y^2+z^2+1$, to the sets given by the Piatetski-Shapiro sequences.

Figures (2)

• Figure 1: The graph of a smooth function $x\in[1,\infty)\mapsto f(x)\in\mathbb R$ and an associated slightly curved sequence $\{b_n\}_{n=1}^\infty$: $|b_n-f(n)|=O(1)$$(n\to\infty)$.
• Figure 2: The white and black dots altogether indicate part of the graph of the map $n\in\mathbb N\mapsto a_n(x)\in\mathbb N$ for each $x\in E_{S,A}=R_{t,L}(S_*,A\setminus S_*,(M_k)_{k=1}^\infty,(W_k)_{k=1}^\infty)$. The black dots indicate the digits in $W_k$ (resp. $W_{k+1}$) inserted in between time $M_k$ and time $M_k+1$ (resp. between time $M_{k+1}$ and time $M_{k+1}+1$).

Theorems & Definitions (39)

• Theorem 1.1: fractal transference principle
• Theorem 1.2: cf. BL
• Theorem 1.3
• Theorem 1.4: relative fractal transference principle
• Theorem 1.5: TZ
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8: SP19
• Theorem 1.9
• Theorem 1.10