q-Real Numbers: Quantum Deformation

• q-real numbers are canonical power or Laurent series defined via continued fractions and q-integers, forming a q-deformation of the classical real line.
• They exhibit rigorous algebraic and analytic properties such as strict monotonicity, injectivity, and specific convergence radii, with rational cases expressed as q-rational functions.
• The construction interplays with modular group actions and finds applications in quantum groups, knot theory, and combinatorial models, bridging classical and quantum mathematics.

A $q$-real number, or $q$-deformed real, refers to a canonical power series or Laurent series over $\mathbb{Z}[[q]][q^{-1}]$ associated to each real number, defined through an interplay between continued fractions, $q$-analogues of integers, and the (quantized) action of the modular group via Möbius transformations. This notion provides a bridge between classical and quantum mathematics, offering a highly structured $q$-analogue of the real line with remarkable algebraic, analytic, combinatorial, and symmetry properties.

1. Foundational Definition and Construction

Given $x\in\mathbb{R}$, the $q$-real $[x]_q$ is constructed in two stages:

• For rational $x$: If $x = [a_1, a_2, \ldots, a_{2m}]$ is its (even-length) regular continued fraction, define

$$[x]_q = [a_1]_q + \cfrac{q^{a_1}}{ [a_2]_{q^{-1}} + \cfrac{q^{-a_2}}{ [a_3]_q + \cdots + \cfrac{q^{-a_{2m-1}}}{ [a_{2m}]_{q^{-1}} } } }$$

where $[n]_q = 1 + q + \cdots + q^{n-1}$ is the classical $q$-integer. This yields $[x]_q \in \mathbb{Q}(q)$ as a rational function.

• For irrational $x$: For any sequence $(x_n)$ of rationals converging to $x$, the sequence of power series $[x_n]_q$ stabilizes coefficientwise, defining

$$[x]_q = \lim_{n \to \infty} [x_n]_q \in \mathbb{Z}((q))$$

as a formal power or Laurent series. This limit is independent of the choice of approximants and is guaranteed by the stabilization property for continued fraction convergents. The construction for negative $x$ is extended via explicit $q$-analogues of translation and inversion:

$$[x+1]_q = 1 + q[x]_q, \qquad [x-1]_q = \frac{[x]_q-1}{q}$$

and for $x<0$ by iterated application of backward shifts (Morier-Genoud et al., 2019, Morier-Genoud et al., 31 Mar 2025, Machacek et al., 2023).

This $q$-deformation recovers the identity $\lim_{q \to 1^-} [x]_q = x$, yielding a flat classical limit.

2. Algebraic and Analytic Properties

• Valuation and Order: The smallest exponent of $q$ in $[x]_q$ reflects the integral part of $x$. If $x\ge1$, $\operatorname{ord}([x]_q)=0$; for $0$\operatorname{ord}([x]_q)\ge1$; and for $x<0$, the order equals (the lower integer bounding $x$).
• Monotonicity and Injectivity: The map $x\mapsto [x]_q$ is strictly increasing in the lexicographic order on coefficients for $0Jouteur, 3 Mar 2025). For $q\in\mathbb{R}_{>0}$, $[x]_q$ is continuous and strictly increasing as a function of $x$ with respect to the coefficientwise order (Machacek et al., 2023).
• Addition and Forward Difference: Translation by integers is encoded as

$$[x+n]_q = [n]_q + q^n[x]_q,\qquad [x-n]_q = q^{-n}([x]_q - [n]_q)$$

and the forward-difference $\{x\}_q = [x+1]_q - [x]_q = 1 + (q-1)[x]_q$ plays the role of a $q$-exponential shift.

• Radius of Convergence: For $q\in\mathbb{C}$, the series $[x]_q$ has a finite radius of convergence $R(x)$. The "Hurwitz-type" conjecture asserts $R(x) \geq (3-\sqrt{5})/2$ for all $x>0$, with equality only for $x$ in the $PSL(2,\mathbb{Z})$-orbit of the golden ratio, and this lower bound is verified in extensive classes (Leclere et al., 2021, Ren, 2021).
• Analytic Structure: For $x\in\mathbb{Q}$, $[x]_q$ is a rational function. For $x$ irrational, $[x]_q$ is a formal power series (or Laurent series for $x<0$), and is algebraic over $\mathbb{Z}(q)$ only for quadratic irrationals (satisfying explicit $q$-quadratic equations with palindromic discriminant).

3. Symmetry and Modular Group Action

A cornerstone is the compatibility of $q$-real numbers with the $q$-deformed action of $PSL(2,\mathbb{Z})$:

• Action: The modular group acts on the classical real projective line by M\"obius transformations and on $[x]_q$ via explicit $q$-deformed matrix substitutions:

$$[M\cdot x]_q = M_q \cdot [x]_q \qquad (M\in PSL(2,\mathbb{Z}))$$

where $M_q$ is the corresponding matrix in $PGL_2(\mathbb{Z}[q, q^{-1}])$ (Leclere et al., 2021, Jouteur, 3 Mar 2025).

• Extended Symmetries: The group $PGL_2(\mathbb{Z}) \times \mathbb{Z}/2$ extends to quantized involutions and twisted actions, relating left and right versions of $q$-reals and encoding further algebraic symmetries (Jouteur, 3 Mar 2025).

4. Examples and Explicit Formulas

Table: Classical vs. $q$-Real Expansions

Classical Number	Continued Fraction	$q$-Real Expansion
$n$ (integer)	$[n]$	$[n]_q = 1 + q + \cdots + q^{n-1}$
$\frac{5}{3}$	$[1,1,2]$	$\frac{1+2q+2q^2+q^3}{1+q+q^2}$
$(1+\sqrt{5})/2$	$[1,1,1,\ldots]$	$1 + q^2 - q^3 + 2q^4 -4q^5 + \ldots$
$1+\sqrt{2}$	$[2,2,2,\ldots]$	$1+q+q^4-2q^6+q^7+\cdots$

Quadratic irrationals yield $q$-reals satisfying algebraic equations: for the golden ratio,

$qG^2 + (1-q-q^2)G = 1$

where $G = [(1+\sqrt{5})/2]_q$, and the discriminant is a palindromic polynomial. For metallic numbers $[n]_q$, the associated $q$-real satisfies a quadratic with palindromic coefficients (Ren, 2021, Machacek et al., 2023).

5. Metric, Combinatorial, and Fractal Properties

• Total Positivity: For $x > y$, the difference $[x]_q - [y]_q$ has positive integer coefficients (Morier-Genoud et al., 31 Mar 2025, Machacek et al., 2023).
• Palindromicity and Unimodality: For $x\in\mathbb{Q}$, the numerator and denominator of $[x]_q$ are palindromic and unimodal polynomials in $q$ (Leclere et al., 2021, Machacek et al., 2023).
• Combinatorial Models: The coefficient of $q^k$ in $[x]_q$ counts north-east lattice paths in snake graphs or order ideals in specific posets arising from the continued fraction of $x$. For $q = p^\ell$, these coefficients count certain $\mathbb{F}_q$-points in Grassmannian varieties (Machacek et al., 2023, Morier-Genoud et al., 31 Mar 2025).
• Somos and Gale–Robinson Sequences: Hankel determinants built from the coefficients of special $q$-reals such as the $q$-golden ratio are periodic and take only the values $-1,0,1$, satisfying bilinear recurrences of Somos/Gale–Robinson type (Ovsienko et al., 2023).

6. Connections, Applications, and Significance

$q$-reals are deeply linked to numerous mathematical domains:

• Quantum Groups and Representation Theory: The $q$-integers and continued fraction phenomena underpin the structure of $U_q(\mathfrak{sl}_2)$, quantum planes, and quantum Teichmüller spaces (Morier-Genoud et al., 31 Mar 2025).
• Knot Theory: The numerator/denominator polynomials of $q$-rationals coincide with the Jones polynomials of two-bridge knots, and certain companion polynomials are obtained directly from the $q$-deformation construction.
• Cluster Algebras and Combinatorics: The snake graph combinatorics, unimodality, and Hankel determinant periodicity index close ties to cluster algebras of type A and integrable systems (Ovsienko et al., 2023).
• Dynamics and Fractal Geometry: The $q$-continued fractions lead to multifractal phenomena in Hausdorff dimensions of certain digit sets, with applications to expansions in non-integer bases and unique expansion problems (Baker et al., 2021, Vries et al., 2021).

The $q$-real numbers provide a rich algebraic, analytic, and combinatorial structure interpolating between classical and quantum mathematics, yielding robust applications and bridging disparate mathematical subfields through the unifying language of $q$-deformation.