q-Real Numbers: Modular Deformations

• q-Real numbers are formal power series with integer coefficients that encode classical real numbers via a q-deformation process and modular invariance.
• They utilize q-integers and q-continued fractions to translate classical arithmetic into combinatorial structures with palindromic and unimodal properties.
• The stabilization of power series coefficients under rational approximations underpins their analytic, combinatorial, and topological applications in quantum groups and fractal geometry.

A $q$-real number is a formal mathematical object—a rational function or power series in the variable $q$ with integer coefficients—that encodes a classical real number (rational or irrational) in such a way that setting $q = 1$ recovers the original number. The defining principle of $q$-real numbers is invariance under the modular group action: the assignment $x \mapsto [x]_q$ intertwines the action of $PSL(2,\mathbb{Z})$ on the real projective line and an explicit $q$-deformation of its linear fractional transformations. This construction, developed in the works of Morier-Genoud and Ovsienko and furthered by other mathematicians, is deeply connected to combinatorics, representation theory, quantum groups, knot theory, discrete integrable systems, and fractal geometry (Morier-Genoud et al., 31 Mar 2025, Morier-Genoud et al., 2020, Morier-Genoud et al., 2018, Leclere et al., 2021).

1. Foundational Construction and Modular Invariance

The $q$-real number $[x]_q$ assigned to $x \in \mathbb{R}$ is uniquely characterized by a geometric compatibility with the modular group $PSL(2, \mathbb{Z})$. Specifically, for each $A \in PSL(2, \mathbb{Z})$ acting as $x \mapsto (ax + b)/(cx + d)$, there exists a $q$-deformation $A_q$ acting on $q$-real numbers (by $q$-linear fractional transformation), such that

$[A(x)]_q = A_q([x]_q).$

This property uniquely determines $[x]_q$ once a basepoint is chosen, usually $[0]_q = 0$ (Morier-Genoud et al., 31 Mar 2025, Leclere et al., 2021). The modular invariance extends to $PGL_2(\mathbb{Z}) \times \mathbb{Z}_2$ via a twisted action, and for rational numbers, there are two natural "right" and "left" versions of $q$-deformation corresponding to distinct choices of basepoints, but for irrationals, the distinction fades as both constructions converge to the same series (Jouteur, 3 Mar 2025).

2. $q$-Integers, $q$-Continued Fractions, and $q$-Rationals

At the algebraic level, the basic building block is the $q$-integer,

$$[n]_q = 1 + q + q^2 + \cdots + q^{n-1} = \frac{1-q^n}{1-q}.$$

This generalizes to rational numbers via $q$-deformed continued fraction expansions: $$[n/m]_q = [c_0]_q - \frac{q^{c_0-1}}{[c_1]_q - \frac{q^{c_1-1}}{[c_2]_q - \cdots}},$$ where $[c_i]_q$ are $q$-integers and the powers of $q$ encode the combinatorial complexity of the expansion (Morier-Genoud et al., 31 Mar 2025, Leclere et al., 2021, Morier-Genoud et al., 2018). For any rational $x$, one computes $[x]_q$ via a sequence of modular actions starting from $0$ or $1/0$. The resulting $q$-rational is a quotient of two palindromic, totally positive polynomials with integer coefficients. These polynomials often have deep combinatorial significance, counting objects such as subrepresentations of quivers, lattice paths, or tilings; their unimodality and palindromicity are observed and partially proved (Morier-Genoud et al., 2018).

3. $q$-Deformed Irrationals and Stabilization Phenomenon

For irrational $x \in \mathbb{R}$, the $q$-real number $[x]_q$ is obtained as the limit of $q$-rational approximations constructed from the rational convergents of $x$: $[x]_q := \lim_{n \to \infty} [x_n]_q,$ where $x_n \in \mathbb{Q}$ converge to $x$. The power series expansion of $[x]_q$ stabilizes term-by-term as $n$ increases; for each fixed $k$, the coefficient of $q^k$ in $[x_n]_q$ ceases to change after finitely many terms, a phenomenon proved for both rational and irrational $x$ (Morier-Genoud et al., 2019, Leclere et al., 2021, Morier-Genoud et al., 2020, Morier-Genoud et al., 31 Mar 2025). This stabilization is essential in defining $q$-real numbers as analytic objects in the ring $\mathbb{Z}[[q]]$, and is central to their combinatorial and analytic properties.

4. Algebraic and Analytical Properties

Combinatorics and Positivity

• The numerator and denominator polynomials of $q$-rational numbers are totally positive, palindromic, and often unimodal (Morier-Genoud et al., 2018, Leclere et al., 2021).
• These polynomials coincide (up to sign and shift) with F-polynomials of cluster algebras in certain cases and, for specific sequences (Fibonacci, Pell), with generalized Catalan numbers or other integer sequences.

Binomial and Gamma Functions

• Generalized $q$-binomial coefficients and $q$-Gamma functions are constructed from $q$-rational and $q$-real numbers, replacing the ill-defined $q^\alpha$ exponent with

$\{\alpha\}_q = [\alpha+1]_q - [\alpha]_q,$

which serves as an analogue of $q^\alpha$ for arbitrary real $\alpha$ (Machacek et al., 2023).

Recurrences and Hankel Determinants

• Hankel determinants of the coefficients of $q$-real numbers often obey discrete integrable recurrences (Somos-4, Gale-Robinson) and display periodicity with values in $\{-1,0,1\}$ (Ovsienko et al., 2023).

Radius of Convergence

• The analytic properties of $q$-real numbers depend on the radius of convergence of their power series expansions; sharp lower bounds and extremality results are established, notably that the $q$-deformed golden ratio has the minimal radius of convergence among all $q$-real numbers (Leclere et al., 2021, Ren, 2021).

5. Fractal, Metric, and Topological Aspects

A parallel branch of $q$-real number theory is the paper of $q$-expansions—expansions of $x$ in non-integer base $q$ using a specified alphabet—leading to fractal structures: $$x = \sum_{i=1}^\infty d_i q^{-i}, \quad d_i \in \{0,1,\dotsc,M\}.$$ Key sets include the univoque set $\mathcal{U}_q$ (numbers with unique $q$-expansion) and sets $\mathcal{U}_q^{(k)}$ of numbers with precisely $k$ different expansions (Dajani et al., 2015, Baker et al., 2021, Vries et al., 2021).

• For $q$ below a critical value $q_c$ (the root of $x^3-3x^2+2x-1=0$), the univoque set is trivial.
• For $q > q_c$, spaces of unique and multiple expansions coexist with intricate metric and topological character.
• Hausdorff dimension computations reveal discontinuity phenomena and fractal complexity:

$$\dim_H \mathcal{U}_q^{(k)} = \dim_H \mathcal{U}_q, \quad \forall k \geq 2, \; q \in (q_c, \infty)$$

and for continuum multiplicity, the dimension is full (Dajani et al., 2015).

• Combinatorial structure is characterized via lexicographic conditions on digit sequences, and shift spaces of finite type are employed to encode the set of unique expansions (Vries et al., 2021).

6. Applications and Connections to Other Areas

$q$-real numbers link several disparate domains:

• Quantum Groups/Quantum Calculus: $q$-numbers are prototypical in the definition of quantum algebras and $q$-binomial formulas.
• Knot Theory: The Jones polynomial of rational knots is expressible in terms of $q$-continuant polynomials derived from $q$-rationals (Morier-Genoud et al., 2018, Morier-Genoud et al., 2020).
• Cluster Algebras: Numerators/denominators of $q$-rational numbers coincide with cluster F-polynomials in certain quiver settings (Morier-Genoud et al., 2018).
• Discrete Geometry/Combinatorics: Snake graph tilings, Farey graph mediants, and P-partitions furnish direct combinatorial interpretations for the coefficients in $q$-rational polynomials (Burcroff et al., 13 Aug 2024).
• Algebraic Geometry: The trace polynomials of $q$-deformed modular elements are palindromic and are connected to hyperbolic geometry and modular forms (Leclere et al., 2021, Morier-Genoud et al., 31 Mar 2025).
• Mathematical Physics: The deformation process provides a formal quantization of the real line, with consequences for quantized invariants and integrable models (Morier-Genoud et al., 31 Mar 2025).

7. Advanced Generalizations and Future Directions

Recent advances include "higher $q$-continued fractions," which generalize classical $q$-rational numbers via ratios of generating functions for P-partitions on posets, with matrix formulas capturing the combinatorial core (Burcroff et al., 13 Aug 2024). This framework preserves the stabilization phenomenon and positivity properties seen in the core theory.

A plausible implication is that $q$-real numbers—by encoding arithmetic, geometric, and combinatorial data—will continue to inform research in quantum topology, number theory, algebraic geometry (modular/automorphic forms), and the theory of fractals. The injectivity of the quantization process and the modular invariance set the stage for further categorical or representation-theoretic interpretations (Jouteur, 3 Mar 2025).

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In summary, $q$-real numbers unite modular-invariant quantization procedures, combinatorial structures, and analytic properties to form a robust framework encompassing $q$-deformations of integers, rationals, irrationals, and real numbers via continued fractions, modular symmetries, and stabilization limits. Their paper bridges diverse areas ranging from quantum algebra to fractal geometry.