Rational q Classification: Structures & Models

• Rational q classification is the study of rational numbers organized by a rational parameter q, yielding distinct arithmetic and geometric structures through D(q)-tuples, q-deformations, and generalized expansions.
• Methodologies include constructing elliptic curves, applying continued fraction theory, and leveraging combinatorial models like fence posets and snake graphs to classify invariants.
• Applications span density theorems, dynamical shift operators, and finite-field orbit counts, with significant implications and open conjectures driving current research.

Searching arXiv for recent and foundational papers on classifications involving rational numbers, $q$-rationals, and $D(q)$-structures. “Rational number $q$ classification” denotes a cluster of classification problems in which either a rational parameter $q$ determines the arithmetic or geometric structure under study, or a $q$-deformation assigns a structured invariant to an ordinary rational number. In the literature represented here, this phrase covers several technically distinct but conceptually related settings: rational $D(q)$-tuples and their density questions, $q$-deformed rational numbers defined from continued fractions, rationality criteria for generalized $q$-ary and Cantor-series expansions, decomposition of rationals by continued-fraction complexity, and finite-field classification of cubic rational expressions over $\mathbb{F}_q$. Across these settings, the central theme is that rational numbers are not treated merely as isolated values, but as objects organized by equivalence relations, recurrence structures, continued fractions, elliptic curves, combinatorial models, or arithmetic congruence data (Dražić, 2021).

1. Parameter-dependent rational structures

A first major usage of $q$ in classification theory treats $D(q)$0 as a fixed nonzero rational parameter and asks which rational configurations exist for that value. The basic example is the notion of a rational $D(q)$1-$D(q)$2-tuple: a set of distinct nonzero rationals

$D(q)$3

such that $D(q)$4 is a square in $D(q)$5 for every $D(q)$6 (Dražić, 2021). This definition generalizes the classical $D(q)$7-problem and makes the existence theory explicitly $D(q)$8-dependent.

For quintuples, the main conditional classification result is density-theoretic. Assuming the Parity Conjecture for eight explicitly constructed quadratic-twist families $D(q)$9, the set of squarefree integers $q$0 for which at least one twist has root number $q$1 contains at least $q$2 out of the $q$3 residue classes mod $q$4 that can contain squarefree $q$5. Equivalently, the density of squarefree $q$6 admitting infinitely many rational $q$7-quintuples is at least

$q$8

(Dražić, 2021). The argument proceeds by constructing a universal elliptic curve $q$9, extracting eight points whose associated polynomials $q$0 have degree $q$1 or $q$2, reducing the existence problem to the quadratic-twist equation $q$3, and then using periodicity of root numbers together with the Parity Conjecture (Dražić, 2021).

A complementary classification appears for rational $q$4-quadruples with prescribed product $q$5. For fixed $q$6, there exists a rational $q$7-quadruple of product $q$8 if and only if

$q$9

for some $q$0 (Dražić et al., 2020). This is an exact existence criterion rather than a density statement. The proof passes through the genus-$q$1 curve

$q$2

and its birational Weierstrass model

$q$3

after which all rational $q$4-quadruples of product $q$5 are recovered from triples of rational points on $q$6 satisfying a non-degeneracy condition and a square-class condition (Dražić et al., 2020).

These two results illustrate two different modes of $q$7-classification. One classifies which $q$8 are “good” for the existence of infinitely many objects; the other classifies which auxiliary invariants $q$9 occur once $D(q)$0 is fixed.

2. $D(q)$1-deformed rational numbers

A second major direction takes an ordinary positive rational number $D(q)$2 and assigns to it a rational function in $D(q)$3, usually denoted $D(q)$4. In the formulation of Morier-Genoud and Ovsienko, $D(q)$5-deformed rationals are defined from continued fractions and encoded by coprime polynomials $D(q)$6 with

$D(q)$7

(Morier-Genoud et al., 2018). The construction is compatible with both regular and negative continued fractions, and Theorem 1 of that work states that the two resulting $D(q)$8-deformed continued fractions coincide (Morier-Genoud et al., 2018).

This framework supports several parallel classification schemes. One is matrix-theoretic: $D(q)$9-analogues of convergent matrices and $q$0-continuants recover the numerator and denominator polynomials, with

$q$1

for a negative continued fraction $q$2 (Morier-Genoud et al., 2018). Another is combinatorial: the coefficients of $q$3 and $q$4 count closure sets, quiver subrepresentations, and objects attached to a triangulation or Farey-graph recursion (Morier-Genoud et al., 2018). A plausible implication is that $q$5-rationality here is less a deformation of arithmetic operations than a reorganization of continued-fraction data into algebraic and enumerative invariants.

Later work sharpened the arithmetic classification of the denominator polynomial $q$6. If $q$7, then $q$8; more strikingly, if $q$9, then again

$q$0

(Kogiso et al., 2024). For prime $q$1, computational evidence up to $q$2 supports the converse: equality of denominator polynomials should occur only from these two congruence patterns (Kogiso et al., 2024). The same paper also gives root-of-unity criteria: $q$3 where $q$4 (Kogiso et al., 2024). This converts the classification of $q$5-denominators into an arithmetic orbit problem governed by congruence mod $q$6, inverse-mod symmetry, and cyclotomic divisibility.

3. Combinatorial and geometric models for $q$7-rationals

The classification of $q$8-deformed rationals is not confined to continued fractions. Several papers recast the same objects in combinatorial, geometric, and categorical terms.

One such model uses hyperbinary partitions. Let $q$9 be the length-generating polynomial for hyperbinary partitions of $\mathbb{F}_q$0, with recurrences

$\mathbb{F}_q$1

and $\mathbb{F}_q$2 (McConville et al., 27 Aug 2025). If $\mathbb{F}_q$3 occurs as the $\mathbb{F}_q$4th entry in the Calkin–Wilf enumeration of the non-negative rationals, then

$\mathbb{F}_q$5

(McConville et al., 27 Aug 2025). The same paper identifies a fence poset $\mathbb{F}_q$6 whose order-ideal lattice is isomorphic to the poset of hyperbinary partitions of $\mathbb{F}_q$7 ordered by refinement, and also expresses the relevant matrix products in terms of the same polynomials $\mathbb{F}_q$8. This yields a unified classification by Calkin–Wilf position, fence-poset rank generating functions, and $\mathbb{F}_q$9-matrix products (McConville et al., 27 Aug 2025).

A closely related but broader combinatorial account is given by a classification in terms of fence posets, snake graphs, and intervals in Young’s lattice. For a continued fraction $q$0, the numerator polynomial $q$1 of $q$2 is the rank-generating function of the lattice of order ideals in a fence poset $q$3, equivalently the generating function of paths in a snake graph $q$4, and also

$q$5

for a pair of partitions $q$6 determined by the snake graph (Ovenhouse, 2021). The same numerator counts the $q$7-points of a union of open Schubert cells in a Grassmannian: $q$8 (Ovenhouse, 2021). Here the pair $q$9 classifies the numerator polynomial, while the continued-fraction data classify the underlying skew shape.

More recent work extends these interpretations to all positive rationals and uses a single combinatorial object for both numerator and denominator. For an even-length continued fraction $D(q)$00, the polynomials $D(q)$01 and $D(q)$02 arise simultaneously from a $D(q)$03-Ostrowski numeration system, order ideals of a fence poset, perfect matchings of a snake graph, and integer points in a convex polytope cut by an affine hyperplane (Aval et al., 14 Nov 2025). This suggests that the classification of $D(q)$04-rationals is best understood as a web of equivalences among continued fractions, admissible sequences, distributive lattices, perfect matchings, and polyhedral combinatorics.

A topological-categorical version appears in the construction of bigraded simple closed arcs $D(q)$05 on a decorated disk. In that setting,

$D(q)$06

and the same data categorify to spherical objects $D(q)$07 in a Calabi–Yau–$D(q)$08 category of type $D(q)$09 (Fan et al., 2023). The map $D(q)$10 is a bijection from $D(q)$11 to the set of simple closed arcs in the decorated disk, so rational numbers themselves are classified by topological isotopy classes carrying $D(q)$12-intersection invariants (Fan et al., 2023).

4. Rationality criteria in generalized $D(q)$13-ary and Cantor-series expansions

A different sense of rational-number classification arises when one fixes a sequence $D(q)$14 of integers $D(q)$15 and asks which real numbers in $D(q)$16 are rational in terms of their $D(q)$17-Cantor expansions

$D(q)$18

with $D(q)$19 (Serbenyuk, 2017).

The fundamental theorem is the shift-periodicity criterion: if

$D(q)$20

then

$D(q)$21

where $D(q)$22 is the Cantor-series shift operator (Serbenyuk, 2017). In other words, rationality is exactly eventual periodicity of the shifted tail. The survey also records the finite-expansion test, the Diananda–Oppenheim condensation criterion, and an eventual constant-digit-ratio criterion under mild growth hypotheses (Serbenyuk, 2017).

A related but more operator-theoretic classification uses generalized shift operators $D(q)$23 obtained by deleting the $D(q)$24th digit from a positive Cantor-series expansion. Writing

$D(q)$25

Theorem 1 states that

$D(q)$26

(Serbenyuk, 2021). Under the uniqueness assumption excluding an all-$D(q)$27 tail, a second theorem gives a digit-wise recurrence criterion for rationality, in which each digit $D(q)$28 is forced by a linear rule depending on a rational value $D(q)$29, the radices $D(q)$30, and the prefix data $D(q)$31 (Serbenyuk, 2021).

These results classify rational numbers not by value alone but by dynamical properties of digit-expansion operators. In the constant-radix case $D(q)$32, they reduce to the classical statement that a base-$D(q)$33 expansion is rational if and only if it is eventually periodic (Serbenyuk, 2021). In the variable-radix setting, the same principle survives but is encoded by nonuniform shift maps.

5. Continued-fraction complexity and decomposition of rationals

A further classification principle organizes a reduced rational $D(q)$34 by the complexity of the continued fractions appearing in a decomposition

$D(q)$35

(Bourgain, 2012). Bourgain proved that there exists an absolute constant $D(q)$36 such that every reduced rational $D(q)$37 admits such a finite decomposition with

$D(q)$38

where $D(q)$39 are the partial quotients of the regular continued fraction of $D(q)$40 (Bourgain, 2012).

The proof uses a peeling-off algorithm. One chooses a bound $D(q)$41 with $D(q)$42 admissible, invokes zero-density results toward Zaremba’s conjecture to find many denominators admitting a numerator whose partial quotients are all at most $D(q)$43, imposes congruence conditions through Proposition $D(q)$44, and then repeatedly replaces a given fraction by one with uniformly bounded partial quotients plus remainder terms of smaller denominator-size (Bourgain, 2012). After at most $D(q)$45 steps, every term has denominator at most $D(q)$46, so the procedure terminates (Bourgain, 2012).

This does not classify rationals by a finite normal form in the strict algebraic sense, but it classifies them by an asymptotic complexity invariant: every height-$D(q)$47 rational admits a decomposition whose total continued-fraction complexity is $D(q)$48, and the paper states that the $D(q)$49 growth is essentially optimal since any continued fraction of denominator $D(q)$50 has $D(q)$51 (Bourgain, 2012).

6. Finite-field classification of cubic rational expressions over $D(q)$52

In a distinct but standard number-theoretic usage, $D(q)$53 denotes the size of a finite field. The paper on cubic rational expressions studies degree-$D(q)$54 rational maps

$D(q)$55

over $D(q)$56, up to pre- and post-composition by independent Möbius transformations, i.e. under the action of

$D(q)$57

(Mattarei et al., 2021). Two rational expressions are equivalent if they lie in the same $D(q)$58-orbit. The principal invariants are ramification points, ramification indices, and branch points (Mattarei et al., 2021).

The classification is complete in even characteristic. When $D(q)$59 is even, every cubic has at most two ramification points, and there are exactly $D(q)$60 equivalence classes if $D(q)$61 is a square, and $D(q)$62 otherwise (Mattarei et al., 2021). Canonical representatives include $D(q)$63, $D(q)$64, $D(q)$65, and several parametrized families with prescribed ramification data (Mattarei et al., 2021).

For odd characteristic $D(q)$66, the few-ramification case is classified completely: every cubic with at most three ramification points is equivalent to one of

$D(q)$67

with $D(q)$68 a nonsquare in $D(q)$69 in the relevant cases (Mattarei et al., 2021). In the generic four-point case, the paper proves an upper bound of $D(q)$70 for the number of $D(q)$71-orbits (Mattarei et al., 2021). The argument uses the total count $D(q)$72 of cubic expressions together with stabilizer analysis: over $D(q)$73, the stabilizer is either the Klein four or, when the ramification cross-ratio is a primitive sixth root of unity, isomorphic to $D(q)$74 (Mattarei et al., 2021).

This finite-field problem belongs to the same broad family of $D(q)$75-classifications only in notation: $D(q)$76 is no longer a rational parameter or deformation variable, but the field cardinality governing orbit counts, ramification configurations, and the size of equivalence classes. The conceptual parallel is that in all cases $D(q)$77 controls the taxonomy of rational objects.

7. Scope, limitations, and common patterns

The studies above do not define a single universal notion of “rational number $D(q)$78 classification.” Rather, they exhibit several mathematically precise classification programs in which $D(q)$79 plays one of three roles.

First, $D(q)$80 may be an arithmetic parameter defining an existence problem, as in rational $D(q)$81-tuples and the elliptic curves $D(q)$82 or $D(q)$83 (Dražić, 2021). Second, $D(q)$84 may be a deformation variable producing polynomial or rational-function invariants of an ordinary rational $D(q)$85, classified by continued fractions, congruence symmetries, fence posets, snake graphs, Schubert-cell unions, or categorical intersection data (Morier-Genoud et al., 2018). Third, $D(q)$86 may denote a radix sequence or field size, so that rationality or equivalence is classified by shift periodicity, digit recurrences, or group-orbit structure over $D(q)$87 (Serbenyuk, 2021).

A common misconception is to treat all of these as variants of the same $D(q)$88-analogue. The data do not support that identification. The $D(q)$89-literature concerns a parameter in Diophantine conditions; the $D(q)$90-rational literature concerns deformation of positive rationals via continued fractions; the Cantor-series literature concerns rationality criteria in generalized expansions; and the finite-field paper concerns cubic rational maps over a field of size $D(q)$91. What unifies them is methodological rather than definitional: each turns rational objects into families indexed by $D(q)$92, then classifies those families by explicit invariants.

A plausible implication is that the most robust classification mechanisms are those that admit multiple equivalent realizations. For $D(q)$93-deformed rationals, continued fractions, matrices, fence posets, snake graphs, Young-lattice intervals, Schubert-cell counts, topological arcs, and categorical dimensions all encode the same object (Ovenhouse, 2021). For $D(q)$94-structures, elliptic curves serve an analogous unifying role, converting existence and parametrization into rank, twist, and square-class questions (Dražić et al., 2020). For generalized expansions, shift operators turn rationality into a recurrence or periodicity problem (Serbenyuk, 2017).

Several limitations remain explicit. The quintuple density theorem is conditional on the Parity Conjecture (Dražić, 2021). The arithmetic converse for denominator-polynomial coincidences is conjectural for prime $D(q)$95 and fails in composite cases (Kogiso et al., 2024). In finite fields of odd characteristic, the generic four-ramification-point cubic case admits only an upper bound of $D(q)$96 rather than a full list of normal forms (Mattarei et al., 2021). These limitations indicate that $D(q)$97-classification, in its various forms, is an active area in which exact classification is often available only for special regimes, while generic regimes are treated by bounds, density results, or conjectural correspondences.