Beatty solutions of almost Golomb equations

Abstract: The Golomb equation of order~$r$ is the implicit functional equation $$a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$$ for nondecreasing sequences of positive integers. Its earliest solution, the almost Golomb sequence of order~$r$, is $r$-regular in the sense of Allouche and Shallit and has oscillating ratio $a(n)/n$. We prove that for every $r\ge 2$ that is not an even perfect square, the equation admits a second monotone solution given by an inhomogeneous Beatty sequence.

• The paper demonstrates that for all r ≥ 2 (excluding even perfect squares) the almost Golomb equation admits a Beatty solution with a(n)/n converging to 1/√r.
• The paper employs rigorous combinatorial and analytic methods—including continuous functional equations and sawtooth discretization—to precisely characterize the solution behaviors.
• The paper reveals that numerical analyses distinguish Beatty solutions from r-regular greedy solutions, linking these findings to Sturmian words and morphic sequence structures.

Beatty Solutions for Almost Golomb Equations: Characterizations and Structural Properties

Introduction and Background

The paper "Beatty solutions of almost Golomb equations" (2604.10822) investigates monotone solutions of a generalization of Golomb's sequence defined by a sliding-window functional equation. Specifically, Golomb's original sequence is extended from its classic cumulative formulation to a finite-memory variant of order $r$:

$a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$

for a nondecreasing sequence $a(n)$ with $a(k)=0$ for $k<1$. The greedy (earliest) solution is $r$-regular and characterized by oscillatory behavior in its ratio $a(n)/n$. This paper establishes the existence of a second class of monotone solutions given by inhomogeneous Beatty sequences, parameterized by irrational slopes, and analyzes their coexistence, structural distinctions, and implications for discrete functional equations.

Characterization of Beatty Solutions

The main result demonstrates that for all $r\ge 2$ not an even perfect square, the almost Golomb equation admits a monotone Beatty solution:

$$a(n)=\left\lfloor \frac{n}{\sqrt{r}} + \frac{\sqrt{r}}{2} \right\rfloor$$

such that

$a\left(\sum_{j=0}^{r-1}a(n-j)\right) = n$

for all $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$0. For even perfect squares $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$1, the left side equals $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$2 systematically, introducing a defect.

The greedy solution, by contrast, is always $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$3-regular, constructed via minimal consistent growth, and exhibits persistent oscillations in $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$4 that do not converge. The Beatty solution features asymptotic convergence: $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$5, with tightly bounded residuals corresponding to Sturmian words.

A rigorous combinatorial and analytic argument establishes the validity of the Beatty solution for $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$6 not an even perfect square, utilizing a continuous functional equation and sawtooth discretization. For $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$7, Hermite's identity enables an exact analysis of the discrete window sum, revealing sharp phase transitions based on the parity of $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$8.

Numerical and Structural Analysis

Strong numerical results confirm the coexistence and divergence of Monotone solutions:

• For $a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$9, the two solutions coincide for $a(n)$0 and diverge at $a(n)$1, with the Beatty solution "jumping ahead" while the greedy repeats.
• The residuals $a(n)$2 are strictly bounded, corresponding to Sturmian remainder sequences.
• Oscillatory behavior in the greedy solution is persistent and unbounded, contrasted with convergence in the Beatty solution.
• Density and gap distribution analysis show that defect positions (where the strong identity fails but the triple-nested identity holds) are characterized by rotation interval conditions and exhibit morphic (possibly substitutive) structure.

The paper provides explicit frequency computations and rigorous bounds for small non-square values of $a(n)$3, verifying the inequalities necessary for the Beatty solution by direct evaluation.

Triple-Nested Equations and Beatty Interval

A triple-nested version of the equation,

$a(n)$4

admits a continuum of Beatty solutions, parameterized by a shift $a(n)$5 within a sharp interval $a(n)$6. Below the lower bound and above the upper bound, the identity fails at infinitely many $a(n)$7, which is elucidated via continued fraction convergents and Pell equation data.

The endpoints of this interval match the parameters of Wythoff complementary Beatty partitions and the classical $a(n)$8-Wythoff game, showing deep connections to combinatorial game theory and algebraic properties. The unique shift yielding both the strong identity and Beatty complementation coincides with $a(n)$9.

Sturmian and Ostrowski Structure

The Beatty solution's difference sequence forms a Sturmian word with slope $a(k)=0$0 and intercept derived via continued fractions and Ostrowski numeration. The multiplicity function aligns with the structure of the Sturmian rotation, and morphic generation is possible through canonical substitutions.

For $a(k)=0$1, the full solution space of monotone sequences forms a binary tree indexed by $a(k)=0$2, with the Beatty solution lying outside the tree. The paper references a companion work [Cloitre_tree] for detailed classification of monotone solutions and their regularity.

Defect Analysis and Morphic Conjecture

At the right endpoint $a(k)=0$3, the strong identity fails at positions determined by inclusion of the rotation address in an explicit interval. The density of defect positions is $a(k)=0$4, and gaps between defects take on three values. Empirical evidence suggests that this gap sequence is morphic, possibly governed by a primitive substitution, though a precise description is conjectural.

Grouping gap values into two classes (short/long) yields a sequence with algebraic frequencies matching those from Sturmian theory.

Open Questions and Future Directions

The paper poses several technical open problems:

• Uniqueness of the affine solution for the continuous equation, subject to homeomorphic monotonicity.
• Classification of monotone solutions for $a(k)=0$5, with only the binary tree structure completely described for $a(k)=0$6.
• Carry-free interpretation in Ostrowski numeration, relating digit swaps to the identity for the Beatty solution.
• Rigorous identification of the morphic structure governing defect sets for both $a(k)=0$7 and general $a(k)=0$8.

Implications of the research stretch to combinatorics on words, automatic sequences, and game theory, particularly regarding explicit enumeration and partitioning of integer sets through functional equations with self-referential addressing.

Conclusion

This paper provides a comprehensive characterization of Beatty solutions for almost Golomb equations, demonstrating their existence, structure, and distinction from $a(k)=0$9-regular greedy solutions, except for even perfect squares. The work connects discrete self-referential functional equations to irrational rotations, combinatorial game partitions, Sturmian words, and Ostrowski numeration systems, unveiling a rich interplay among algebraic, combinatorial, and analytic properties. Future work may resolve structural conjectures, extend classification to higher orders, and deepen the relationship to automatic and morphic sequences.