Perfect Set of Liouville Numbers

• The perfect set of Liouville numbers is defined as a closed, uncountable subset with no isolated points, where every element is exceptionally well-approximable by rationals.
• Cantor-type constructions are used to ensure that finite sums, products, self-powers, and pairwise exponentiations of elements remain within the Liouville universe.
• These sets, despite having Lebesgue measure zero, exhibit rich algebraic closure and fractal properties, linking transcendental number theory with topology.

A perfect set of Liouville numbers is a closed, uncountable subset of the real line with no isolated points, in which every element is a Liouville number—a real number that is “exceptionally” well-approximable by rationals according to Liouville’s criterion. Recent work demonstrates the existence of such perfect sets with remarkable algebraic closure properties: not only are finite sums and products of elements again Liouville, but also their self-powers and, under refinement, all pairwise exponentiations remain Liouville. The perfect set thus supports complex algebraic and transcendental operations entirely within the Liouville universe, while retaining topological largeness: cardinality continuum and the perfect-set property (Morris et al., 21 Nov 2025, Álvarez-Samaniego et al., 2016).

1. Liouville Numbers: Definition and Topological Properties

A real number $x$ is called a Liouville number if for every integer $N \ge 1$ there exist $p \in \mathbb{Z}$ and $q \in \mathbb{N}$, $q \ge 2$, such that $|x - p/q| < q^{-N}$. The set of all Liouville numbers, denoted $\mathcal{L}$, satisfies the following classical properties:

• $\mathcal{L}$ is uncountable and has cardinality continuum.
• $\mathcal{L}$ is a dense $G_\delta$ subset of $\mathbb{R}$.
• The Lebesgue measure of $\mathcal{L}$ is zero.
• $\mathcal{L}$ contains no isolated points (Morris et al., 21 Nov 2025, Álvarez-Samaniego et al., 2016).

These facts establish that $\mathcal{L}$ is topologically large (dense and uncountable) but measure-theoretically negligible.

2. Perfect Sets and Cantor-Type Constructions

A subset $P \subset \mathbb{R}$ is called perfect if it is closed, uncountable, and contains no isolated points. Classical Cantor sets are paradigmatic perfect sets. Bendixson’s theorem asserts that any closed subset of $\mathbb{R}$ splits uniquely into a perfect set and a countable remainder; every nonempty perfect subset contains a Cantor set (Álvarez-Samaniego et al., 2016).

Explicit Cantor-type constructions inside $\mathcal{L}$ are possible. For example, the “factorial-digit” Cantor set $C_L$ defined as

$$C_L = \left\{ y = \sum_{n=1}^\infty \frac{x_{n-1}}{10^{n!}} : (x_n) \in A \right\},$$

where $A \subset \{0,1\}^{\mathbb{N}}$ ensures for all $k \ge 0$ that $x_{2k} + x_{2k+1} = 1$, yields a closed, perfect, nowhere dense (thus Cantor) subset $C_L \subset \mathcal{L}$ of cardinality continuum. This construction leverages the super-exponential decay of the series tails to guarantee the Liouville property for all elements (Álvarez-Samaniego et al., 2016).

In (Morris et al., 21 Nov 2025), a Cantor–type set

$$S = \left\{ x_a = \sum_{n=1}^\infty \frac{a_n}{3^{e_n}} : a_n \in \{0,2\},\, a_n = 2\, \text{infinitely often} \right\}$$

with $e_1 = 3$, $e_{n+1} = 3^{e_n}$, is constructed and shown to be perfect and of continuum cardinality. Again, the super-exponential rate ensures that every $x_a \in S$ is a Liouville number.

3. Algebraic Closure and Polynomial Maps

An innovation in (Morris et al., 21 Nov 2025) is the algebraic closure under polynomial maps. For a perfect set $P \subset \mathcal{L}$, for every finite set $\{x_1, \dots, x_t\} \subset P$ and nonconstant $Q \in \mathbb{Z}[X_1, \dots, X_t]$,

$$Q(x_1, \dots, x_t) \in \mathbb{Q} \quad \text{or} \quad Q(x_1, \dots, x_t) \in \mathcal{L}.$$

That is, finite sums, finite products, and general polynomial expressions (excluding elements mapping to $\mathbb{Q}$) remain in $\mathcal{L}$.

The proof uses truncations $x_j^{(m)}$ of elements in $S$, rational approximations with rapidly increasing denominators, and polynomial mean-value estimates to show that, whenever the polynomial value is not rational, it admits infinitely many rational approximations of arbitrarily high accuracy, thus guaranteeing the Liouville property for the result.

4. Self-Powers and Pairwise-Power Closure

A central advance is the construction of perfect subsets closed under the transformation $x \mapsto x^x$. Given $x \in (0,1)$, write $u = x \log x$, and $x^x = e^{u}$. Using Lipschitz continuity of $u(x)$ on compact subintervals and by inductively defining subsets where $u$ admits extremely precise rational approximations, a Cantor-type subset $T \subset S$ is built with the property that every $x \in T$, $x^x$ is Liouville.

This process exploits the controlled growth of denominators in rational Taylor approximations of $e^{A/B}$, yielding for every large $n$ a rational $\tilde{P}_n/\tilde{Q}_n$ such that $|x^x - \tilde{P}_n/\tilde{Q}_n| < \tilde{Q}_n^{-N}$ for arbitrary $N$ (Morris et al., 21 Nov 2025).

Further refinement isolates a perfect subset $W \subset T$ so that all pairwise exponentiations $x^y$ for $x, y \in W$ also remain Liouville. The argument constructs sequences where exponents $y \log x$ approach required rational anchors, facilitating Liouville-approximation for $x^y$ via the same machinery.

5. Structural and Cardinality Results

Each construction (sets $S$, $T$, $W$) is realized as an intersection of nested families of finitely many closed “cylinders” inside the base Cantor set, with digit constraints at each stage and infinitely many degrees of freedom remaining after each construction step. Consequently, the final set is closed, perfect, nonempty, and of cardinality $2^{\aleph_0}$.

The procedure avoids isolated points at each finite stage, ensuring perfectness. The intersection across countably many stages, each with binary freedom, secures continuum cardinality. The resulting sets are nowhere dense due to containment in the measure-zero Liouville set (Morris et al., 21 Nov 2025, Álvarez-Samaniego et al., 2016).

6. Implications for Fractal and Descriptive Topology

The existence of perfect sets and Cantor subsets within the Liouville set signals deep interplay between transcendental number theory and the topology of fractal sets. Although $\mathcal{L}$ has measure zero and lacks interior, it supports algebraic and transcendental closure properties typically associated with much larger algebraic structures.

From the descriptive-topology perspective, the Liouville set is a dense $G_\delta$ with fractal substructure, and perfect sets manifest the coexistence of transcendentally rich and topologically complex families within the reals. The necessary and sufficient condition for a subset of $\mathbb{R}$ to contain a Cantor set is simply the existence of a closed uncountable subset (Álvarez-Samaniego et al., 2016). A plausible implication is that similar closure phenomena may be tractable in other exceptional sets of reals.

7. Related Constructions in Diophantine Sets

Analogous Cantor-type sets exist for Diophantine numbers—the complement of Liouville numbers in $\mathbb{R}$—with corresponding closure and density properties. Both the $\mathcal{L}$ and $\mathcal{D}$ sets are dense, the former $G_\delta$ of measure zero, the latter $F_\sigma$ of full measure. This suggests that transcendence (via Liouville’s criterion) and fractal perfectness are orthogonal: perfect sets abound in both regimes (Álvarez-Samaniego et al., 2016).

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These results collectively elucidate the fine-grained algebraic and topological richness of the set of Liouville numbers, demonstrating that topologically large perfect subsets can be constructed with robust closure properties under addition, multiplication, and transcendental maps such as self-powers and pairwise exponentiation (Morris et al., 21 Nov 2025, Álvarez-Samaniego et al., 2016).