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

$q$0

where $q$1 is the classical $q$2-integer. This yields $q$3 as a rational function.

• For irrational $q$4: For any sequence $q$5 of rationals converging to $q$6, the sequence of power series $q$7 stabilizes coefficientwise, defining

$q$8

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 $q$9 is extended via explicit $\mathbb{Z}[[q]][q^{-1}]$0-analogues of translation and inversion:

$\mathbb{Z}[[q]][q^{-1}]$1

and for $\mathbb{Z}[[q]][q^{-1}]$2 by iterated application of backward shifts (Morier-Genoud et al., 2019, Morier-Genoud et al., 31 Mar 2025, Machacek et al., 2023).

This $\mathbb{Z}[[q]][q^{-1}]$3-deformation recovers the identity $\mathbb{Z}[[q]][q^{-1}]$4, yielding a flat classical limit.

2. Algebraic and Analytic Properties

• Valuation and Order: The smallest exponent of $\mathbb{Z}[[q]][q^{-1}]$5 in $\mathbb{Z}[[q]][q^{-1}]$6 reflects the integral part of $\mathbb{Z}[[q]][q^{-1}]$7. If $\mathbb{Z}[[q]][q^{-1}]$8, $\mathbb{Z}[[q]][q^{-1}]$9; for $q$0, $q$1; and for $q$2, the order equals (the lower integer bounding $q$3).
• Monotonicity and Injectivity: The map $q$4 is strictly increasing in the lexicographic order on coefficients for $q$5, and is injective (Jouteur, 3 Mar 2025). For $q$6, $q$7 is continuous and strictly increasing as a function of $q$8 with respect to the coefficientwise order (Machacek et al., 2023).
• Addition and Forward Difference: Translation by integers is encoded as

$q$9

and the forward-difference $q$0 plays the role of a $q$1-exponential shift.

• Radius of Convergence: For $q$2, the series $q$3 has a finite radius of convergence $q$4. The "Hurwitz-type" conjecture asserts $q$5 for all $q$6, with equality only for $q$7 in the $q$8-orbit of the golden ratio, and this lower bound is verified in extensive classes (Leclere et al., 2021, Ren, 2021).
• Analytic Structure: For $q$9, $x\in\mathbb{R}$0 is a rational function. For $x\in\mathbb{R}$1 irrational, $x\in\mathbb{R}$2 is a formal power series (or Laurent series for $x\in\mathbb{R}$3), and is algebraic over $x\in\mathbb{R}$4 only for quadratic irrationals (satisfying explicit $x\in\mathbb{R}$5-quadratic equations with palindromic discriminant).

3. Symmetry and Modular Group Action

A cornerstone is the compatibility of $x\in\mathbb{R}$6-real numbers with the $x\in\mathbb{R}$7-deformed action of $x\in\mathbb{R}$8:

• Action: The modular group acts on the classical real projective line by M\"obius transformations and on $x\in\mathbb{R}$9 via explicit $q$0-deformed matrix substitutions:

$q$1

where $q$2 is the corresponding matrix in $q$3 (Leclere et al., 2021, Jouteur, 3 Mar 2025).

• Extended Symmetries: The group $q$4 extends to quantized involutions and twisted actions, relating left and right versions of $q$5-reals and encoding further algebraic symmetries (Jouteur, 3 Mar 2025).

4. Examples and Explicit Formulas

Table: Classical vs. $q$6-Real Expansions

Classical Number	Continued Fraction	$q$7-Real Expansion
$q$8 (integer)	$q$9	$[x]_q$0
$[x]_q$1	$[x]_q$2	$[x]_q$3
$[x]_q$4	$[x]_q$5	$[x]_q$6
$[x]_q$7	$[x]_q$8	$[x]_q$9

Quadratic irrationals yield $x$0-reals satisfying algebraic equations: for the golden ratio,

$x$1

where $x$2, and the discriminant is a palindromic polynomial. For metallic numbers $x$3, the associated $x$4-real satisfies a quadratic with palindromic coefficients (Ren, 2021, Machacek et al., 2023).

5. Metric, Combinatorial, and Fractal Properties

• Total Positivity: For $x$5, the difference $x$6 has positive integer coefficients (Morier-Genoud et al., 31 Mar 2025, Machacek et al., 2023).
• Palindromicity and Unimodality: For $x$7, the numerator and denominator of $x$8 are palindromic and unimodal polynomials in $x$9 (Leclere et al., 2021, Machacek et al., 2023).
• Combinatorial Models: The coefficient of $x = [a_1, a_2, \ldots, a_{2m}]$0 in $x = [a_1, a_2, \ldots, a_{2m}]$1 counts north-east lattice paths in snake graphs or order ideals in specific posets arising from the continued fraction of $x = [a_1, a_2, \ldots, a_{2m}]$2. For $x = [a_1, a_2, \ldots, a_{2m}]$3, these coefficients count certain $x = [a_1, a_2, \ldots, a_{2m}]$4-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 $x = [a_1, a_2, \ldots, a_{2m}]$5-reals such as the $x = [a_1, a_2, \ldots, a_{2m}]$6-golden ratio are periodic and take only the values $x = [a_1, a_2, \ldots, a_{2m}]$7, satisfying bilinear recurrences of Somos/Gale–Robinson type (Ovsienko et al., 2023).

6. Connections, Applications, and Significance

$x = [a_1, a_2, \ldots, a_{2m}]$8-reals are deeply linked to numerous mathematical domains:

• Quantum Groups and Representation Theory: The $x = [a_1, a_2, \ldots, a_{2m}]$9-integers and continued fraction phenomena underpin the structure of $q$00, quantum planes, and quantum Teichmüller spaces (Morier-Genoud et al., 31 Mar 2025).
• Knot Theory: The numerator/denominator polynomials of $q$01-rationals coincide with the Jones polynomials of two-bridge knots, and certain companion polynomials are obtained directly from the $q$02-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$03-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$04-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$05-deformation.