---
title: Balanced Quasi-Arcs in Domains & Self-Similarity
url: https://www.emergentmind.com/topics/balanced-quasi-arcs
type: topic
---

# Balanced Quasi-Arcs in Domains & Self-Similarity

Searching arXiv for papers on balanced quasi-metrics/formal balls and quasi-arcs to ground the article.
Balanced quasi-arcs is not a standard formal term in the cited literature. In the available arXiv sources, it is most naturally understood as an interpretive umbrella for two adjacent structures: first, the balanced quasi-metric \(q_b\) on the domain of finite and infinite words, together with its formal-ball domain; second, \(t\)-quasi-arcs among self-similar arcs in \(\mathbb{R}^N\). Read in combination, these works suggest a notion of arc-like directed approximation that is “balanced” either in the quasi-metric sense of Doitchinov or in the geometric sense of uniform control across scales and vertices [1607.05298, 1703.10665].

## 1. Terminological scope and ambient settings

The phrase *balanced quasi-arcs* does not appear as a formal definition in either source. One source studies a balanced quasi-metric on a domain of words and proves that its poset of formal balls is a continuous domain; the other studies strict Whitney arcs and \(t\)-quasi self-similar arcs, obtaining local and global criteria for quasi-arc behavior. A plausible implication is that the phrase can be used to describe arc-like structures whose directionality, approximation, and regularity are controlled by one of these two frameworks [1607.05298, 1703.10665].

In the word-domain setting, the ambient space is \(\Sigma^\infty\), the set of finite and infinite words over a non-empty alphabet \(\Sigma\), ordered by prefix:
\[
x \sqsubseteq y \quad\text{iff}\quad x \text{ is a prefix of } y.
\]
This order makes \((\Sigma^\infty,\sqsubseteq)\) a domain in the sense of domain theory. In the geometric setting, the ambient space is an arc \(\Lambda\subset \mathbb{R}^N\), typically self-similar, with subarcs \(\Lambda(x,y)\) and Euclidean distance \(|x-y|\). The two settings are structurally different, but both organize approximation by directed refinement: longer prefixes approximate an infinite word, and smaller self-similar pieces approximate a limiting arc.

A common misconception is to treat these two notions of “balance” as identical. They are not. In the first setting, balance refers to a quasi-metric property yielding a Hausdorff and completely regular induced topology. In the second, balance is suggested by geometric control such as positive corner angles or, in degenerate cases, by arithmetic constraints on scaling parameters. The papers support comparison, but they do not collapse the two terminologies into a single established theory.

## 2. Balanced quasi-metric structure on the domain of words

The balanced quasi-metric studied on \(\Sigma^\infty\) is
\[
q_b(x,y) =
\begin{cases}
2^{-\ell(x)} - 2^{-\ell(y)}, & \text{if } x \text{ is a prefix of } y,\\[4pt]
1, & \text{otherwise.}
\end{cases}
\]
Here \(\ell(x)\) is the length of \(x\), with \(\ell(\varnothing)=0\) and \(\ell(x)=\infty\) for infinite words. The empty word is denoted by \(\varnothing\), and \(x\wedge y\) denotes the longest common prefix of \(x\) and \(y\) [1607.05298].

This quasi-metric is directional. If \(x \prec y \prec z\), then
\[
q_b(y,z) < q_b(x,z),
\]
so a longer prefix is closer to the target word in the quasi-metric sense. Prefix detection is encoded sharply: for \(x\neq \varnothing\), \(x\) is a prefix of \(y\) if and only if \(q_b(x,y)<1\); otherwise the distance is exactly \(1\). The space is \(T_1\), and in fact \(q_b(x,y)=0\Rightarrow x=y\). The paper states that \(q_b\) is a balanced quasi-metric in the sense of Doitchinov, and that its induced topology is Hausdorff and completely regular.

For comparison, the classical Baire metric is
\[
d_B(x,x)=0,\qquad d_B(x,y)=2^{-\ell(x\wedge y)}\quad\text{if }x\neq y.
\]
Because \(d_B\) is symmetric, it does not distinguish the prefix relation directionally. The quasi-metric \(q_b\) does, and this directional asymmetry is the key feature that makes it natural for modeling progressively better approximations.

This suggests an arc-like interpretation: increasing prefix chains behave like discrete directed paths. A left \(K\)-Cauchy sequence in \((\Sigma^\infty,q_b)\) is eventually non-decreasing under \(\sqsubseteq\), hence eventually constant or convergent to an infinite word obtained as the union of a strictly increasing prefix chain. In this setting, “quasi-arc” is not a formal term from the paper, but the behavior is explicitly path-like.

## 3. Formal balls and the continuous-domain model of approximation

For a quasi-metric space \((X,d)\), a formal ball is a pair \((x,r)\in X\times \mathbb{R}_+\). The order on formal balls is
\[
(x,r)\le_d (y,s)\quad\Longleftrightarrow\quad d(x,y)\le r-s.
\]
Applied to \((\Sigma^\infty,q_b)\), this yields the poset
\[
B\Sigma^\infty=\Sigma^\infty\times \mathbb{R}_+,
\qquad
(x,r)\le_{q_b}(y,s)\Longleftrightarrow q_b(x,y)\le r-s.
\]
The paper proves that \((\Sigma^\infty,q_b)\) is Yoneda-complete and therefore that \((B\Sigma^\infty,\le_{q_b})\) is a dcpo. Its main structural theorem is stronger: the poset of formal balls is a domain, i.e. a continuous dcpo [1607.05298].

Continuity is expressed through the way-below relation. For finite words \(x\) and all \(u\in\mathbb{R}_+\), \(v>0\),
\[
(x,u+v)\ll_{q_b}(x,u).
\]
Thus a formal ball centered at a finite word is the supremum of strictly coarser balls with the same center and slightly larger radii. For an infinite word \(x\), if \((x_n)\) is a chain of finite prefixes with \(\ell(x_n)=n\), \(x_n\prec x_{n+1}\), and \(x_n\sqsubseteq x\), then
\[
(x_n,r+2^{-n})
\]
is an ascending chain in \((B\Sigma^\infty,\le_{q_b})\) with supremum \((x,r)\), and each term is way-below \((x,r)\).

These constructions furnish canonical directed approximation chains. For a finite center, one has thickened approximants such as
\[
(x,r+1),\ (x,r+\tfrac12),\ (x,r+\tfrac14),\dots
\]
For an infinite center, one has prefix-based approximants
\[
(x_1,r+2^{-1})\le_{q_b}(x_2,r+2^{-2})\le_{q_b}\cdots.
\]
The paper does not call these objects arcs, but it explicitly describes them as directed and convergent in the domain-theoretic sense. A plausible implication is that the formal-ball domain supplies a natural ambient space for “balanced quasi-arcs” understood as Scott-continuous or directed approximation paths.

## 4. \(t\)-quasi-arcs and strict Whitney structure in self-similar geometry

In the geometric literature, an arc \(\Lambda\) is a homeomorphic image of \([0,1]\) in \(\mathbb{R}^N\). For \(x,y\in \Lambda\), let \(\Lambda(x,y)\) denote the subarc between them and let \(|\Lambda(x,y)|\) be its diameter. For \(t\ge 1\), \(\Lambda\) is a \(t\)-quasi-arc if there exists \(\lambda>0\) such that
\[
|\Lambda(x,y)|^t \le \lambda\,|x-y|,\qquad \forall x,y\in\Lambda.
\]
A \(1\)-quasi-arc is called a quasi-arc. The inequality cannot hold for \(t<1\) because \(|\Lambda(x,y)|\ge |x-y|\) [1703.10665].

The paper studies self-similar arcs generated by finitely many contractive similitudes \(S_1,\dots,S_\ell\), with Hausdorff dimension \(s\) determined by
\[
\sum_{j=1}^\ell r_j^s=1.
\]
A self-similar arc is defined by the conditions that \(S_i(\Lambda)\cap S_j(\Lambda)\) is a singleton when \(|i-j|=1\) and empty when \(|i-j|>1\). Such an arc is non-trivial when \(s>1\).

The same paper places quasi-arc theory beside Whitney theory. A connected set \(E\subset\mathbb{R}^N\) is a strict Whitney set if there exists a real-valued \(C^1\) function \(f\) with \(\nabla f|_E\equiv 0\) such that \(f\) is constant on no non-empty relatively open subsets of \(E\). Its criticality is
\[
\Cr(E)=\sup\left\{ r : \exists\ \text{non-constant } f:E\to\mathbb{R},\, \exists M>0 \text{ s.t. } |f(x)-f(y)|\le M|x-y|^r,\ \forall x,y\in E\right\}.
\]
For a self-similar arc \(\Lambda\) of Hausdorff dimension \(s>1\), the paper proves
\[
\Cr(\Lambda)=s,
\]
and states that \(\Lambda\) is a strict Whitney set [1703.10665].

The function most relevant to the geometry is the Hausdorff measure function
\[
f(x)=\mathcal{H}^s([z_0,x]),\qquad x\in\Lambda.
\]
It is increasing along the natural order of the arc, and the paper relates its Whitney behavior to quasi-arc behavior through local criteria at vertices. This creates a precise link between analytic flattening, geometric distortion, and self-similar scaling.

## 5. Local vertex conditions, corner angles, and arithmetic obstructions

For a self-similar arc with vertices \(z_0\prec \dots \prec z_\ell\), the paper defines local conditions at each inner vertex \(z_p\). Condition \(W_p\) is
\[
|f(x)-f(y)| = o(|x-y|)
\quad \text{for } z_{p-1}\preceq x\preceq z_p\preceq y\preceq z_{p+1},
\]
and Condition \(Q_p^t\) is the existence of \(C_p>0\) such that
\[
|\Lambda(x,y)|^t \le C_p\,|x-y|
\quad\text{whenever } z_{p-1}\preceq x\preceq z_p\preceq y\preceq z_{p+1}.
\]
The global strict Whitney condition holds if and only if every \(W_p\) holds, and \(\Lambda\) is a \(t\)-quasi-arc if and only if every \(Q_p^t\) holds [1703.10665].

For regular self-similar arcs in \(\mathbb{R}^2\), the geometry is encoded by corner angles
\[
\theta_p=\arg\frac{S_p(z_q)-z_p}{S_{p+1}(z_q)-z_p},\qquad 0\le \theta_p<2\pi,
\]
their minimum
\[
\theta_{\min}=\min\{\theta_1,\dots,\theta_{\ell-1}\},
\]
and the characteristic angle
\[
\xi:=\theta_{\min}+\eta_0.
\]
A self-similar arc generated by a basic figure with \(\xi>0\) is called regular. If \(\theta_p>0\), then the paper proves that \(W_p\) holds and that \(Q_p^t\) holds for every \(t\ge 1\). Consequently, if
\[
\theta_{\min}>0,
\]
then the arc is a \(t\)-quasi-arc for every \(t\ge 1\), and the \(s\)-dimensional Hausdorff measure function is a Whitney function.

A second misconception is that positive minimal corner angle is necessary for quasi-arc behavior. The paper shows that it is only a sufficient condition. When \(\theta_{\min}=0\), quasi-arc behavior may still occur, but the criterion becomes arithmetic rather than purely geometric. At a flat vertex with \(\theta_p=0\), Condition \(W_p\) is equivalent to
\[
(\lambda^j+\mu^k)^s=o(|\alpha\lambda^j-\beta\mu^k|),\qquad j,k\in\mathbb{Z}_+,
\]
for parameters \(\lambda,\mu,\alpha,\beta\) derived from the local similitudes. Thus zero-angle junctions are the locus of subtle behavior rather than automatic failure.

## 6. One-parameter families and the synthesized notion of balanced quasi-arcs

The paper constructs a one-parameter family \(\{\Lambda_\tau\}_{1<\tau<1.001}\) of regular self-similar arcs in \(\mathbb{R}^2\) for which exactly one corner has angle \(0\), namely \(\theta_3=0\), while the other corner angles are positive. In this family,
\[
\zeta=\log\frac{15}{7},\qquad x=\zeta,\qquad y=\frac{\zeta}{\tau},
\]
and the \(t\)-quasi-arc property for irrational \(\tau\) is equivalent to the Diophantine approximation property \(J_{(t-1)\zeta}\):
\[
|\tau-k/j|\ge Cj^{-1}e^{-aj}\quad \forall k\in\mathbb{Z},\ j\in\mathbb{N},
\]
with \(a=(t-1)\zeta\). Concretely,
\[
\Lambda_\tau \text{ is a } t\text{-quasi-arc}
\quad\Longleftrightarrow\quad
\tau \text{ has property } J_{(t-1)\zeta},
\qquad t>1.
\]
The paper then proves a range of existence results: for each \(t_0>1\), there are regular self-similar arcs with \(\theta_{\min}=0\) that are \(t_0\)-quasi-arcs but not \(t\)-quasi-arcs for any \(t\in[1,t_0)\); for each \(t_0\ge 1\), there are arcs that are \(t\)-quasi-arcs for all \(t>t_0\) but not \(t_0\)-quasi-arcs; and there exists a regular self-similar arc with \(\theta_{\min}=0\) that is a \(t\)-quasi-arc for no \(t\ge 1\) [1703.10665].

Taken together with the formal-ball results for \(q_b\), these examples suggest a useful synthesized reading of *balanced quasi-arcs*. In the word-domain setting, balance is built into the quasi-metric and continuity of the formal-ball dcpo: every target is the supremum of a directed family of approximants. In the self-similar setting, balance is expressed by the persistence of the quasi-arc inequality across all vertices and scales, either through the geometric condition \(\theta_{\min}>0\) or through arithmetic control at flat vertices. A plausible implication is that a balanced quasi-arc, in the broadest sense supported by these papers, is an arc-like object whose directed approximation structure is globally coherent and whose local singularities do not destroy scale-controlled convergence.

Under that interpretation, the two sources provide complementary models. The balanced quasi-metric on words yields canonical directed chains of formal balls converging to finite or infinite targets. The theory of \(t\)-quasi self-similar arcs yields sharp geometric and arithmetic criteria for when subarc diameter is uniformly controlled by chord length. What is established in the literature is not a single unified theory under the name *balanced quasi-arcs*, but two rigorous frameworks from which such a theory could naturally be assembled.

Source: https://www.emergentmind.com/topics/balanced-quasi-arcs