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Properties of a random Cantor set with overlaps

This paper investigates the topology and Hausdorff dimension of a random Cantor set with overlaps generated by an iterated function system with the Golden Mean scaling ratio, extending known formulas to cases where the Open Set Condition fails by utilizing the theory of expansions in non-integer bases.

Original authors: Anna Chiara Lai, Paola Loreti

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anna Chiara Lai, Paola Loreti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure the roughness of a coastline, or the jagged edge of a snowflake. In the world of math, these aren't just messy drawings; they are objects called "fractals." Fractals are shapes that look similar no matter how much you zoom in. A tiny piece of the edge looks just like the whole edge. For decades, mathematicians have had a reliable rulebook for measuring the "dimension" of these shapes—basically, how much space they fill. If a line is one-dimensional and a square is two-dimensional, a fractal might be 1.5 dimensions, living somewhere in between.

Usually, this rulebook works perfectly when the pieces of the fractal don't bump into each other. It's like building a tower with blocks: if every block sits neatly on top of the one below without touching its neighbors, you can easily count how many blocks you have and predict how tall the tower will get. This is a condition mathematicians call the "Open Set Condition." But what happens when the blocks are too big, or the tower is built in a way that the blocks start to squash into each other? The old rulebook breaks down. The pieces overlap, hide behind one another, and the math gets messy. This is the puzzle our paper tackles: trying to measure the size of a fractal when the building blocks are crashing into each other, specifically using a special number called the "Golden Mean" as the rule for how much they shrink.

The Golden Mean and the Squashing Blocks

In this study, Anna Chiara Lai and Paola Loreti investigate a very specific type of fractal called a "random Cantor set." To understand this, picture a game of "keep or discard" played with a line segment. You start with a long stick. You have two magical machines, let's call them Machine 0 and Machine 1. Machine 0 shrinks the stick and moves it to the left; Machine 1 shrinks it and moves it to the right. The amount they shrink is determined by a famous number: the Golden Mean (often written as $\phi$ ), which is roughly 1.618.

Here is the twist: unlike a normal game where the machines work in separate, tidy zones, these two machines are a bit clumsy. Because the Golden Mean is such a specific number, the pieces they create sometimes land right on top of each other. It's like if you tried to place two stickers on a page, but the glue was so sticky that they merged into one blob. In math terms, this is called an "overlap," and it breaks the standard rules for measuring the shape.

To make things even more interesting, the authors add a layer of chance. Every time the machines try to make a new piece, they flip a coin. If it lands on "heads" (with a probability $p$ ), the piece stays. If it's "tails," the piece is thrown away. They repeat this process over and over, creating a pre-fractal that gets more and more detailed. The big question is: as this process goes on forever, how "big" is the final shape? Does it disappear completely? Does it become a solid line? Or does it stay a weird, dusty cloud?

The Three Zones of Discovery

The authors discovered that the answer depends entirely on how often the coin lands on "heads" (the probability $p$ ). They found three distinct zones, or "territories," where the behavior of the fractal changes completely.

Zone 1: The Vanishing Act ( $p$ is between 0 and 0.5)
If you keep the pieces less than half the time, the fractal simply disappears. It's like trying to build a tower with blocks, but you throw away more than half of them every time you add a layer. Eventually, you run out of blocks entirely. The authors prove that in this zone, the final shape is empty. There is nothing left.

Zone 2: The Dusty Cloud ( $p$ is between 0.5 and roughly 0.809)
This is the most surprising part. If you keep the pieces more than half the time, but not too much more, a shape remains. However, it's not a solid line. It's a "dust" of disconnected points. The authors found a precise formula for how "big" this dust is. The size (called the Hausdorff dimension) grows as you increase the probability $p$ . The formula is:
$\text{Dimension} = \frac{\log(2p)}{\log(\phi)}$
This is a generalization of an old formula that only worked when the blocks didn't overlap. Here, even though the blocks are squashing into each other, the math still follows a predictable pattern, just with a different ceiling.

Zone 3: The Solid Line ( $p$ is between roughly 0.809 and 1)
This is where the magic happens. If you keep the pieces more than about 80.9% of the time, the dimension of the shape jumps to exactly 1. In math terms, this means the shape fills a line. It looks like a solid line to a dimension-measuring tool.

But here is the catch: Even though the dimension says "1" (like a line), the shape is not actually a connected line. It is still a dusty collection of broken pieces. It's as if you had a line made of sand where the grains are so close together that a ruler can't tell the difference between "sand" and "solid," but if you looked closely with a microscope, you'd see gaps everywhere. The authors explicitly state that for any probability $p$ greater than 0, the shape has no connected parts. It never becomes a true, solid line, even though its dimension suggests it should be.

Why This Matters

Before this paper, mathematicians had a hard time dealing with these "squashing" fractals. The standard tools assumed the pieces stayed apart. When they overlapped, the formulas gave wrong answers or no answers at all. Lai and Loreti showed that even in this messy, overlapping scenario, there is a hidden order. They proved that you can still calculate the expected size of the shape using a specific formula, provided you account for the "Golden Mean" rules.

They also identified a critical threshold: the number $\phi/2$ (which is about 0.809). Below this number, the shape is a fractal dust with a fractional dimension. Above it, the shape "thinks" it is a line (dimension 1), even though it is still broken. This discovery helps us understand how randomness and overlap interact in complex systems. It suggests that even when things get messy and overlap, nature (or math) often finds a way to settle into a predictable pattern, provided you know which rules to look for.

The paper doesn't just guess these results; it provides a rigorous proof using a mix of probability theory and the study of how numbers are written in non-standard bases (like writing numbers using the Golden Mean instead of 10). They didn't just simulate this on a computer; they derived the exact mathematical formulas that describe the behavior. This work opens the door to understanding other messy, overlapping systems where the old rules no longer apply.

Technical Summary: Properties of a Random Cantor Set with Overlaps

Problem Statement
This paper investigates the topological structure and Hausdorff dimension of a random Cantor set generated by an Iterated Function System (IFS) that violates the Open Set Condition (OSC). Specifically, the authors analyze a one-dimensional system defined by two contractive maps, $f_0(x) = \frac{x}{\phi}$ and $f_1(x) = \frac{x+1}{\phi}$ , where $\phi = \frac{1+\sqrt{5}}{2}$ is the Golden Mean.

In the deterministic setting, this IFS generates a set where distinct binary sequences can map to the same interval (exact overlaps) or where intervals intersect. The authors introduce a randomization process: starting from the interval $I_0 = [0, \phi]$ , at each iteration $n$ , the images of the current intervals under $f_0$ and $f_1$ are retained with probability $p$ and discarded with probability $1-p$ . The resulting random fractal is the limit set $C_p = \bigcap_{n \ge 0} C_p^n$ .

The central challenge addressed is that standard formulas for the dimension of random Cantor sets, which rely on the OSC (ensuring disjointness of images), fail in this scenario. The violation of OSC arises because the similarity dimension $\frac{\log 2}{\log \phi}$ exceeds the dimension of the ambient space (1), leading to inevitable overlaps and "exact overlaps" where different sequences generate identical intervals.

Methodology
The authors' approach diverges from the measure-theoretic methods often used for IFS with overlaps. Instead, they employ a combinatorial methodology rooted in the theory of expansions in non-integer bases (specifically base $\phi$ ).

Equivalence Classes and Greedy Expansions: The authors define an equivalence relation on binary sequences where $c \sim d$ if they generate the same interval $I_c = I_d$ . To handle the redundancy caused by exact overlaps, they select "greedy expansions" as canonical representatives for these equivalence classes. In base $\phi$ , greedy expansions are characterized by the absence of the forbidden word "011".
Combinatorial Counting: The problem is reduced to counting the number of surviving intervals in the random prefractal $C_p^n$ . This requires estimating the expected number of distinct intervals, which depends on the cardinality of the equivalence classes $[c]$ . The authors derive recursive relations for the size of these classes, leveraging the specific substitution rules (e.g., $100 \to 011$ ) that generate overlaps in base $\phi$ .
Percolation and Dimension Estimation: The construction is modeled as a random binary tree. The authors relate the expected Hausdorff dimension to the growth rate of the expected number of surviving intervals ( $E_n$ ). They establish that $E(\dim(C_p)) = \lim_{n \to \infty} \frac{\log E_n}{n \log \phi}$ .

Key Results
The paper establishes a closed-form formula for the expected Hausdorff dimension of the random Cantor set $C_p$ as a function of the retention probability $p$ :

$E(\dim(C_p)) = \begin{cases} 0 & \text{if } p \in [0, 1/2] \\ \frac{\log(2p)}{\log \phi} & \text{if } p \in [1/2, \phi/2] \\ 1 & \text{if } p \in [\phi/2, 1] \end{cases}$

Key findings include:

Extinction Threshold: For $p \le 1/2$ , the set is empty almost surely, consistent with classical percolation theory on binary trees.
New Critical Threshold: A novel threshold $p_d = \phi/2 \approx 0.809$ emerges. For $p > \phi/2$ , the expected dimension saturates at 1 (the dimension of the ambient space).
Topology vs. Dimension: Despite the dimension reaching 1 for $p > \phi/2$ , the set $C_p$ almost surely contains no connected components (intervals). The set is a "dust" that is topologically disconnected but fills the space in a dimensional sense.
Generalization: The result generalizes the known formula for random Cantor sets satisfying the OSC (where the dimension is $\frac{\log(2p)}{\log q}$ for $p \in [1/2, 1]$ and $q \ge 2$ ) to the case where $q = \phi < 2$ and overlaps occur.

Significance and Claims
The authors claim that this work provides a quantitative and qualitative improvement to the modeling of random fractals by addressing systems that do not satisfy the OSC. By utilizing the specific properties of the Golden Mean base, they demonstrate how redundancy in non-integer base expansions can be quantified combinatorially to determine dimension.

The paper posits that the Golden Mean acts as a critical threshold for the cardinality of unique representations in non-integer bases. Consequently, the study suggests a pathway for understanding the morphology of random Cantor sets with scaling ratios $q \in (1, \phi)$ , and potentially extending to $q \in (1, 2)$ . The authors note that while the current study is one-dimensional, the extension to higher dimensions (random Cantor dust) would require distinct analysis, as it cannot be simply modeled as the Cartesian product of unidimensional sets due to the independence of base and height in such products.

The methodology is presented as a combinatorial alternative to the measure-theoretic approaches found in recent literature on IFS with exact overlaps, offering explicit recursive formulas for the expected number of intervals.

The Golden Mean and the Squashing Blocks

The Three Zones of Discovery

Why This Matters

Technical Summary: Properties of a Random Cantor Set with Overlaps

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