Zeckendorf Shift: Numeration and Combinatorics

• Zeckendorf Shift is an operator that appends zeros to Fibonacci numeral representations, enabling unique integer decompositions and index transformations.
• It underpins the construction of Z-Mahler equations and the development of Z-regular sequences through automata-theoretic approaches.
• The shift facilitates index raising in decomposition classes, impacting combinatorial games and extending to generalized numeration systems.

The Zeckendorf Shift is a central algebraic and combinatorial construct arising naturally in the paper of Zeckendorf representations—unique decompositions of integers as sums of non-consecutive Fibonacci numbers—and their generalizations to combinatorial games, numeration systems, automata theory, and arithmetic encodings. Across these domains, the shift concept encapsulates index manipulations in digit strings, the dynamics of decompositional processes, and operator actions in algebraic and automata-theoretic contexts. Technically, it manifests as explicit digit shifts in representations, transformations of sets under recurrence, operator substitutions in Mahler-type equations, and as a structural driver of transformation invariance in games and decomposition lattices.

1. Definition and Structural Basis

The Zeckendorf shift is formally defined in several, closely related contexts:

• Digit sequence shift: In Zeckendorf numeration, each integer $n$ is represented as a binary word (no adjacent ones), expressing $n$ as $\sum_{k} b_k F_k$, where the $F_k$ are (renormalized) Fibonacci numbers. The shift operator, denoted (in various sources) as $\psi$ or $\mathrm{sh}_F$, acts by appending a zero at the end (least significant digit), corresponding to mapping the integer $n$ to $\psi(n)$ where $n$'s Zeckendorf representation $(b_0b_1...b_\ell)$ becomes $(b_0b_1...b_\ell 0)$:

$\psi(n) = [(n)_Z0]_Z,$

where $[\cdot]_Z$ denotes the Zeckendorf-to-decimal mapping (Carton et al., 3 May 2024).

• Explicit formula: For Zeckendorf representations (with digits least significant first), the shift of $n$ is given by

$\mathrm{sh}_F(n) = \lfloor (n+1)\phi \rfloor - 1,$

and, for a double shift,

$$\mathrm{sh}_F^2(n) = \lfloor (n+1)\phi^2 \rfloor - 2,$$

where $\phi = (1+\sqrt{5})/2$ is the golden ratio (Burns, 8 Jul 2025).

• Automata and lattice perspective: Shifts correspond to transformations (such as moving all indices up in representations, or steps along axes in multidimensional Zeckendorf lattices (Borade et al., 2019)), with statistical properties tied to the pattern of gaps or transitions between summands.

This operator is the Zeckendorf-numeration-system analogue of multiplication by $q$ in base-$q$ expansions, playing an analogous role in operator theory, decomposition structure, and automata-theoretic recognizability.

2. Operator-Theoretic Framework: Z-Mahler Equations

The Zeckendorf shift underpins the construction of Z-Mahler equations—functional equations for power series whose coefficients are read off via Zeckendorf numeration (Carton et al., 3 May 2024). The shift operator $\psi$ induces a non-linear action on exponents, replacing the classical Frobenius map $f(x) \mapsto f(x^q)$ (linear in ordinary $q$-Mahler theory) with

$$\Phi\left( \sum_{n \geq 0} f_n x^n \right) = \sum_{n \geq 0} f_n x^{\psi(n)}.$$

• Nonlinearity and linearity defect: Unlike the $q$-ary case, the Zeckendorf shift does not preserve addition: $\psi(m+n)$ may differ from $\psi(m) + \psi(n)$ by a small, bounded defect $\delta(m, n) \in \{-1,0,1\}$, a measurable function which can be tracked by a finite automaton.
• Z-regular sequences: Solutions to isolating Z-Mahler equations (those with leading coefficient $1$) generate Z-regular sequences—sequences that can be recognized by a weighted automaton reading Zeckendorf representations. Reciprocally, Z-regular sequences satisfy some (possibly non-isolating) Z-Mahler equation.

The Zeckendorf shift is thus the algebraic engine driving the recursive structure of solutions, enabling the construction of weighted automata and the formalization of regularity properties analogous to those in classical $q$-automatic theory (Carton et al., 3 May 2024).

3. Shift Dynamics in Combinatorial Decomposition and Games

3.1. Shifts in Decomposition Classes

The Zeckendorf shift $\mathrm{sh}_F$ and its iterates act as index raisers, transferring classes of numbers between sets defined by their initial (lowest or highest) summand in their Zeckendorf expansion. For example (Burns, 8 Jul 2025):

• If $A_k$ denotes the set of integers whose Zeckendorf representation starts with $F_k$, then $\mathrm{sh}_F(A_k) = A_{k+1}$.
• For fixed digit blocks or avoidance sets, repeated application of the shift reduces complex membership problems to base cases (such as sets avoiding $F_2$).

3.2. Lattice and Gap Structures

In higher-dimensional Zeckendorf lattices, shifts correspond to multivariate motions along axes, and the distribution of vector-valued "gaps" (i.e., the differences between successive summands) characterizes the stochastic shift structure. In two dimensions, the limiting distribution of gaps is bivariate geometric:

$$\lim_{n \to \infty} \mathbb{P}((v_1,v_2)) = 2^{-(v_1+v_2)}$$

with the sum $v_1+v_2$ distributing as $(v-1)/2^v$ for $v \geq 2$ (Borade et al., 2019).

4. Shift Phenomena in Zeckendorf-Based Games

Zeckendorf games—a family of combinatorial games where players transform lists of Fibonacci numbers via merging, splitting, and, in some variants, order-preserving or black hole operations—are governed by the transformation invariance and progressivity created by the Zeckendorf shift.

• Original game (unordered): The move structure (combining or splitting Fibonacci numbers) induces a path through game states that mimics the index evolution in the Zeckendorf representation, with the shift reflecting in the systematic advancement toward the unique Zeckendorf decomposition (Baird-Smith et al., 2018, Cusenza et al., 2020).
• Ordered Zeckendorf game: The introduction of ordered adjacency constraints and a switching move creates a "shift" in winning strategy: for almost all $n \leq 25$, Player 1 (not Player 2) can force a win, demonstrating a reversal of game-theoretic dynamics compared to the unordered version. This is a dynamic Zeckendorf Shift at the level of strategic advantage (Bortnovskyi et al., 27 Aug 2025).
• Black Hole and lattice games: The Zeckendorf shift, in the presence of a black hole column (i.e., truncating columns with tokens lost beyond a threshold), induces a modular reduction (congruence relation) on play, with the final configuration representing the starting number modulo a particular Fibonacci number (Cashman et al., 17 Sep 2024).
• Reversed Zeckendorf game: Flipping moves to reverse the original game's process realizes a Zeckendorf shift "in reverse," leading to different classes of winning configurations and new parity-based strategies (Batterman et al., 2023).

5. Zeckendorf Shift in Numeration, Representation Theory, and Automata

The Zeckendorf shift operator is foundational in various representation-theoretic contexts:

• Automatic sequence recognition: The shift is central to Walnut-based and automata-based proofs for properties of Zeckendorf and Chung–Graham representations (Burns, 8 Jul 2025). The presence of a shift operator admits reductions to base cases and creates automata-theoretic invariance across representation classes.
• Density and leading digit laws: In the context of Benford behavior and digit statistics, shift operations underlie equidistribution phenomena: the fraction of numbers where the leading block matches a prescribed Zeckendorf word is invariant under block-wise Zeckendorf shifts. This is expressed in closed form using logarithmic relations involving the golden ratio (Chang et al., 2023, Best et al., 2014).
• Gödel-type encoding: In arithmetic logic, encoding finite sequences as Zeckendorf sums (via carefully spaced indices) replaces exponentiation-dependent schemes, yielding injective, primitive recursive, and $\Delta_0$-definable codes strictly compatible with bounded arithmetics. Here, the Zeckendorf shift supports substitution and concatenation functions fundamental for fixed-point and diagonal lemmas, aligning syntactic substitution with index shifting in the Fibonacci expansion (Rosko, 12 Sep 2025).

6. Generalizations and Extensions

• Generalized Zeckendorf systems: Positive linear recurrence sequences (PLRS) and their Zeckendorf-type representations admit shift operators defined via their recurrence, extending the "append zero" process to a larger class of numeration systems. The combinatorial and automata-theoretic implications (as in definitions of "legal" or "super-legal" decompositions) rely on analogous shift phenomena (Cordwell et al., 2016, Chang, 2020).
• Weak converse and uniqueness in other settings: The Zeckendorf shift is critical in establishing uniqueness (or its weak converse) of expansion schemes in monotone fundamental sequences, as well as in extending to the $p$-adic and real settings, where the direction (append zero or remove leading digit) encodes lex orderings on the sequence and the corresponding expansion (Chang, 2020).
• Compound sequences and tree structures: The classification of Zeckendorf expansions according to blocks (as in the Fibonacci tree or via Beatty sequences) demonstrates that shift operations enable the recursive construction of occurrence and density sequences, arithmetically tied to index manipulation through shifts (Dekking, 2020).

7. Impact and Applications

The Zeckendorf shift, as an operator and as a combinatorial mechanism, is both a unifying principle and a technical tool. It mediates between recurrence structures, paves the way for automatic and regularity proofs, governs game-theoretic transitions, and underlies efficient, definable encoding schemes in arithmetic logic. Its paper has led to new theorems on decomposition uniqueness, probabilistic laws, and constructive algorithmic frameworks for generalized numeration systems and combinatorial games. The general methodology—reducing complex representation questions to base cases via iterative shifts—has been formalized and automated using the Walnut system and related automata-theoretic approaches (Burns, 8 Jul 2025).

In sum, the Zeckendorf shift is the canonical transformation that lies at the heart of Zeckendorf numeration and its applications to algebraic, combinatorial, and logical domains, encoding the progression, invariance, and reducibility properties central to this rich mathematical ecosystem.