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Gap estimates for the spectrum of $m$ -bonacci numbers

This paper establishes explicit lower bounds for the gaps between elements separated by $N$ positions in the ordered spectrum of $m$ -bonacci numbers by combining the combinatorial structure of $m$ -bonacci words with the canonical $m$ -bonacci number system, with specific applications to Fibonacci and Tribonacci cases.

Original authors: Anna Chiara Lai, Paola Loreti

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anna Chiara Lai, Paola Loreti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a vast, endless hallway where the floor tiles are spaced out in a very specific, rhythmic pattern. You can't just walk anywhere; you can only step on the tiles. This hallway represents a "spectrum," a collection of numbers that follow strict rules. In the world of mathematics, these numbers often come from something called "Pisot numbers." Think of a Pisot number as a special kind of ruler that doesn't measure in neat, even inches like a standard ruler, but instead has marks that grow in a wild, exponential way, yet still manage to fit together without ever getting too messy or chaotic.

Now, imagine you are a curious explorer trying to measure the distance between these tiles. Sometimes the gap between two tiles is small, sometimes it's large, but there's a hidden order to it all. Mathematicians have long known that if you look at the gaps between these special numbers, they don't just randomly jump around; they follow a code, a secret language made of patterns. This paper dives deep into that code, specifically for a family of numbers called "m-bonacci" numbers. These are like the famous Fibonacci numbers (where each number is the sum of the two before it), but stretched out to include sums of three, four, or even more previous numbers. The authors are asking a simple but tricky question: If you skip ahead $N$ steps in this hallway, how far have you definitely traveled? They want to find a guaranteed minimum distance, a safety net that says, "No matter where you start, if you take $N$ steps, you will at least go this far."

The Secret Code of the Number Hallway

In this paper, Anna Chiara Lai and Paola Loreti act like detectives solving a mystery about the spacing of these special numbers. They are looking at the "spectrum" of m-bonacci numbers, which is just a fancy way of listing all the possible numbers you can make by adding up powers of a special number $q_m$ (like $1, q_m, q_m^2$ , etc.) using only 0s and 1s as coefficients. When you line these numbers up from smallest to largest, you get a sequence of "tiles." The space between one tile and the next is called a "gap."

The authors discovered that these gaps aren't random. They are dictated by a "word" made of symbols, much like a sentence made of letters. For the famous Fibonacci numbers, this word is the "Fibonacci word," a sequence of 1s and 2s that never lets the same symbol appear twice in a row in a specific way (you never see "22"). For the broader m-bonacci numbers, there is a similar "m-bonacci word" made of symbols from 1 to $m$ . This word acts as a master key: if the word has a "1" at a certain spot, the gap is one size; if it has a "2," the gap is another size, and so on.

The big breakthrough in this paper is a formula that tells you the minimum distance you must cover if you jump $N$ steps forward in this sequence. The authors proved that for any number of steps $N$ , there is a guaranteed lower bound for the distance. They didn't just guess this; they built a mathematical proof that combines two powerful tools:

The m-bonacci expansion: This is a way of writing the number $N$ as a sum of special m-bonacci numbers (similar to how you might write a number in binary using powers of 2, but here using powers of the m-bonacci sequence).
The "balance" of the word: This is a measure of how evenly the symbols (1, 2, 3...) are distributed in the m-bonacci word. The authors use a constant, called $b_m$ , which acts like a "tolerance" or a "wiggle room" factor. It accounts for the fact that while the word is very orderly, it's not perfectly uniform in every tiny chunk.

The Main Finding: A Guaranteed Minimum Jump

The core result, stated as Theorem 1.1, is a mathematical guarantee. The authors show that if you take $N$ steps in the spectrum of m-bonacci numbers, the total distance you travel, $\lambda_{n+N} - \lambda_n$ , is always greater than or equal to $N$ multiplied by a specific constant, $\gamma_{m,N}$ .

Think of $\gamma_{m,N}$ as the "average speed" of your walk, but calculated with extreme precision based on the specific pattern of your $N$ steps. The formula for this constant is clever: it looks at how $N$ is built from m-bonacci numbers (the expansion) and weighs the different possible gap sizes (the values $d_m(j)$ ) by how often they appear in the m-bonacci word, while subtracting a small "penalty" term ( $b_m$ ) to ensure the estimate is always safe and never too optimistic.

The paper explicitly rules out the idea that you could find a sequence of $N$ steps that is shorter than this calculated limit. For example, in the case of the Fibonacci numbers (where $m=2$ ), they show that you can never find two steps that add up to a distance of $2\phi - 2$ (where $\phi$ is the golden ratio). The structure of the Fibonacci word simply forbids the pattern of gaps that would create such a short distance.

Special Cases: Fibonacci and Tribonacci

The authors didn't stop at the general case; they zoomed in on two famous examples to show how their formula works in practice:

The Fibonacci Case ( $m=2$ ): Here, the "word" is made of 1s and 2s. The authors derived a specific lower bound for the distance after $N$ steps. They noted that because the word never contains "22" (two consecutive large gaps), you can't have two big jumps in a row. This forces the average distance to be higher than if the gaps were random.
The Tribonacci Case ( $m=3$ ): Here, the word uses 1s, 2s, and 3s. The authors provided a more complex formula for this case, involving the Tribonacci constant ( $\tau$ ). They showed that even with three different gap sizes, the combinatorial rules of the word still force a strict minimum distance for any $N$ steps.

Why This Matters

The paper concludes by connecting these gap estimates to the "density" of the spectrum. In simple terms, density is a measure of how crowded the numbers are. If the gaps are small, the numbers are crowded; if the gaps are large, they are sparse. The authors show that their new, precise gap estimates are consistent with the known density of these numbers. They prove that as you take more and more steps ( $N$ gets very large), your calculated minimum average distance approaches the theoretical average distance derived from the density.

In essence, Lai and Loreti have provided a new, sharper ruler for measuring these mathematical hallways. They proved that the hidden order of m-bonacci words isn't just a pretty pattern; it acts as a rigid constraint that prevents the numbers from clustering too tightly, no matter how far you look. Their work confirms that the universe of these numbers is structured, predictable, and governed by the elegant rules of combinatorics.

Technical Summary: Gap Estimates for the Spectrum of m-Bonacci Numbers

Problem Statement
This paper investigates the spectral properties of $m$ -bonacci numbers, specifically focusing on the distribution of gaps between elements in their associated spectra. For a real number $q > 1$ , the spectrum $\Lambda(q)$ is defined as the set of real numbers of the form $\sum_{i=0}^n \varepsilon_i q^i$ where $\varepsilon_i \in \{0, 1\}$ . When ordered increasingly as $\Lambda(q) = \{0 < \lambda_1(q) < \lambda_2(q) < \dots\}$ , the sequence of consecutive differences (gaps) $\lambda_{n+1}(q) - \lambda_n(q)$ exhibits specific structural properties.

While previous literature has established that for Pisot numbers (including the $m$ -bonacci numbers $q_m$ , the largest roots of $x^m - x^{m-1} - \dots - x - 1$ ), the set of gap values is finite and the gaps admit uniform frequencies, this work addresses a more refined question: establishing explicit lower bounds for the distance between elements separated by $N$ positions in the ordered spectrum. Specifically, the authors seek to determine a constant $\gamma_{m,N}$ such that $\lambda_{n+N}(q_m) - \lambda_n(q_m) \geq N \gamma_{m,N}$ for all $n \geq 1$ . The goal is to provide a bound that reflects the weighted average of gap values based on the combinatorial structure of the spectrum, rather than relying solely on the minimal possible gap.

Methodology
The authors employ a methodology that synthesizes the combinatorial structure of $m$ -bonacci words with the canonical $m$ -bonacci number system.

Combinatorial Coding of Gaps: The paper utilizes the result by Bugeaud [9] which establishes a direct correspondence between the gaps in $\Lambda(q_m)$ and the digits of the $m$ -bonacci word $v_m$ . The $m$ -bonacci word is the fixed point of the Rauzy substitution $\sigma_m$ (defined by $k \mapsto 1(k+1)$ for $k < m$ and $m \mapsto 1$ ). The gap value $\lambda_{n+1}(q_m) - \lambda_n(q_m)$ is determined by the $n$ -th symbol of $v_m$ .
Balance Properties: The analysis relies on the "balanced" nature of $m$ -bonacci words. A word is $b$ -balanced if the number of occurrences of any letter in two subwords of the same length differs by at most $b$ . The authors utilize known bounds for the balance constant $b_m$ of $m$ -bonacci words to estimate the frequency of specific gap values within any block of $N$ consecutive gaps.
Canonical Expansions: The authors utilize the canonical $m$ -bonacci expansion of an integer $N$ . This representation expresses $N$ as a sum of $m$ -bonacci numbers $F^{(m)}_h$ with binary coefficients, subject to the constraint that no block of $m$ consecutive 1s appears. This expansion allows the authors to decompose the sum of $N$ consecutive gaps into a weighted sum of the possible gap values $d_m(j)$ .

Key Contributions and Results
The primary contribution is the derivation of an explicit lower bound for the $N$ -step gap in the spectrum of $m$ -bonacci numbers.

Main Theorem (Theorem 1.1): The authors prove that for any $m \geq 2$ and $N \in \mathbb{N}$ , there exists a constant $\gamma_{m,N} > 0$ such that:
$\lambda_{n+N}(q_m) - \lambda_n(q_m) \geq N \gamma_{m,N}$
The constant $\gamma_{m,N}$ is explicitly constructed using the canonical $m$ -bonacci expansion of $N$ (denoted by coefficients $x_h$ ) and the balance constant $b_m$ :
$\gamma_{m,N} = \frac{1}{N} \sum_{j=1}^m \left( \sum_{h \geq 0} x_h F^{(m)}_{h-j} - b_m \right) d_m(j)$
Here, $d_m(j)$ represents the possible gap values, and $F^{(m)}_h$ are the $m$ -bonacci numbers.
Specific Cases (Corollaries 3.1 and 3.2):
- Fibonacci Case ( $m=2$ ): The authors derive a specific lower bound involving the golden mean $\phi$ . They show that $\gamma_{2,N}$ converges to $1/\phi + 1/\phi^3$ as $N \to \infty$ , which corresponds to the inverse of the upper density of the spectrum. They also provide a sharpened estimate for $N=2$ .
- Tribonacci Case ( $m=3$ ): A similar explicit bound is derived for the Tribonacci constant $\tau$ , utilizing the specific balance constant $b_3=2$ and the properties of Tribonacci numbers.
Density Analysis: In Section 4, the paper connects these gap estimates to the upper density $D^+(\Lambda(q_m))$ of the spectrum. The authors demonstrate that as $N \to \infty$ , the derived constant $\gamma_{m,N}$ converges to $1/D^+(\Lambda(q_m))$ , confirming that the proposed bound captures the average distribution of gaps rather than just the minimal gap.

Significance and Claims
The paper claims to provide a "finer investigation" of the spectrum structure compared to previous works that focused only on the existence of gaps or their minimal values. By linking the gap estimates directly to the combinatorial properties of $m$ -bonacci words (specifically balance and forbidden subwords) and the arithmetic properties of $m$ -bonacci expansions, the authors offer a method to calculate explicit, non-trivial lower bounds for multi-step gaps.

The significance lies in the ability to interpret $\gamma_{m,N}$ as a weighted average of gap values, dependent on the specific structure of $N$ . The authors note that while trivial bounds exist (based on the smallest gap), their method yields estimates consistent with the asymptotic density of the spectrum. The work is presented as a contribution to the literature on Pisot numbers, gap distributions, and aperiodic order, specifically extending known results from the Fibonacci case to the broader class of $m$ -bonacci numbers. The paper does not propose new applications or experimental validations but rather establishes theoretical bounds within the framework of numeration systems and symbolic dynamics.

The Secret Code of the Number Hallway

The Main Finding: A Guaranteed Minimum Jump

Special Cases: Fibonacci and Tribonacci

Why This Matters

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