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Additive Bases from Primitive Dyck Words: Regular Underapproximations, Motzkin Coding, and Digit Lifting

This paper establishes that every positive even integer can be represented as a sum of at most six primitive Dyck words, with the exception of a finite set of integers (including 46, which requires eight) and the sharp eventual threshold of 848, by leveraging a novel connection between Dyck paths and Motzkin coding to prove digit-lifting theorems and generation bounds.

Original authors: Takayuki Kuriyama

Published 2026-07-28

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Takayuki Kuriyama

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 a detective trying to solve a very specific kind of number puzzle. In the world of mathematics, there is a branch called additive number theory, which asks a simple but tricky question: Can you build every number in a certain group by adding together a few special "building block" numbers? Think of it like a game where you have a limited set of Lego bricks, and you want to know if you can construct every possible tower height using only those bricks. Sometimes, you might need just two bricks; other times, you might need ten. The "order" of the game is the maximum number of bricks you ever need to build any tower.

To play this game, the mathematicians in this story use a very specific set of building blocks. These blocks are numbers that, when written in binary (the computer language of 0s and 1s), look like perfectly balanced parentheses. In math, these are called Dyck words. For example, 1100 is a valid block because if you treat 1 as an "up" step and 0 as a "down" step, the path goes up twice and comes down twice, never dipping below the starting line. The authors focus on a special subset of these called primitive blocks, which are the "atomic" pieces that can't be broken down into smaller balanced pairs. The big question they tackle is: What is the maximum number of these primitive blocks you need to add together to create any even number?

This paper is a masterclass in solving that puzzle by mixing two different mathematical tools. The authors discovered that these binary blocks have a secret relationship with a different kind of path called a Motzkin path, which allows them to translate the problem into a different language (base-4) where it becomes much easier to solve. They proved that while most even numbers can be built with just a handful of these blocks, there is a small, stubborn group of numbers that are much harder to build. Specifically, they found that the number 46 is the hardest case, requiring eight blocks, while a few others need seven. However, they also proved that once you get past the number 848, you will never need more than six blocks to build any even number, no matter how large. It's a story of finding the "worst-case scenarios" in a vast universe of numbers and proving exactly where the chaos ends and the order begins.

The Story of the Binary Balancers

Let's dive into the adventure. The authors, led by Takayuki Kuriyama, are investigating a set of numbers that come from a language of balanced binary strings. Imagine you have a string of lights, some red (1) and some blue (0). A "Dyck word" is a string where you have the same number of red and blue lights, and if you count them from left to right, you never have more blues than reds at any point. It's like a dance where you can't step off the stage until you've matched every step up with a step down.

The authors are interested in the "primitive" dancers. These are the strings that only return to the starting line (zero height) at the very end. If a string returns to zero halfway through, it's just two smaller dances glued together, not a primitive one. They treat these strings as numbers (reading them as binary) and ask: How many of these primitive numbers do we need to add up to get any even number?

The Secret Code: From Binary to Base-4
The brilliant move in this paper is realizing that these binary strings have a hidden structure. If you pair up the bits (00, 01, 10, 11), they act like digits in a base-4 system (0, 1, 2, 3). The authors found a perfect map: every primitive Dyck number (except for the smallest one, which is 2) corresponds to a base-4 number that starts with a 3, ends with a 0, and has a "Motzkin" word in the middle.

Think of a Motzkin word as a path that can go up, down, or stay flat, but never goes below the ground. This connection is the paper's "Rosetta Stone." It allows the authors to translate a hard problem about complex binary strings into a cleaner problem about base-4 numbers and these flat-walking paths. This translation reveals that the set of numbers they are studying is "digitally closed," meaning if you have a number in the set, you can often generate new ones by tacking on specific digits.

The Two-Track Strategy
To solve the puzzle, the authors use a clever two-pronged attack, treating even numbers based on how they behave when divided by 4.

The "Easy" Track (Multiples of 4): For numbers that are perfectly divisible by 4, the authors use a "regular underapproximation." This is a fancy way of saying they found a simpler, predictable subset of the numbers that is easy to work with. They proved that this simpler set is powerful enough to build all large multiples of 4 using just six blocks.
The "Tricky" Track (Numbers that are 2 mod 4): For numbers that leave a remainder of 2 when divided by 4 (like 6, 10, 14), the simpler set isn't enough. Here, they use the full power of the "Motzkin-coded" family. They proved that this larger, more complex family can build these numbers using just five blocks.

The "Lifting" Magic
How do they know this works for all large numbers, not just the ones they checked? They use a technique called digit lifting. Imagine you have a small ladder that can reach a certain height. The authors proved a theorem that says: if you can build a continuous range of numbers with a certain number of blocks, you can "lift" that ability to build all larger numbers by simply adding specific digits to the ends of the blocks. It's like having a magic rule that says, "If you can build a tower of height 100, you can automatically build towers of height 400, 401, 402, and so on." This allows them to take a finite list of verified numbers and prove that the pattern holds forever.

The Results: The Stubborn Numbers
After setting up their tools, the authors went to work classifying the exceptions. They found that while most even numbers are easy to build, there is a specific list of "stubborn" numbers that require more than six blocks.

The Champion of Difficulty: The number 46 is the hardest of all. It cannot be built with seven or fewer blocks; it strictly requires eight.
The Runners-Up: There are ten other numbers that need seven blocks: 34, 44, 98, 154, 198, 202, 206, 838, 842, and 846.
The Threshold: The authors proved that 848 is the magic number. Every even number from 848 and up can be built with six or fewer blocks.

They didn't just guess these numbers; they used exact computer calculations to verify every single case up to the threshold and used their mathematical proofs to show it holds for infinity.

Why This Matters
This paper is a beautiful example of how different areas of math—computer science (languages and automata), combinatorics (paths and trees), and number theory (addition)—can dance together. The authors didn't just find a list of numbers; they built a framework. They showed that even for a set of numbers defined by a complex, non-repeating pattern (a "context-free" language), you can find a simple, repeating pattern (a "regular" language) that covers most of the ground, and then use the full complexity to fill in the gaps.

They also discovered that the "order" of the game changes depending on the rules. If you only look at multiples of 4, you only ever need 5 blocks. But if you include the numbers that are 2 mod 4, the requirement jumps to 6. And if you look at the absolute worst-case scenario (including the number 46), you need 8.

In the end, the paper gives us a complete map. We know exactly which numbers are the troublemakers, we know the exact limit where the trouble stops, and we have a constructive algorithm (a step-by-step recipe) to build any large even number using these special binary blocks. It turns a chaotic-looking problem into a perfectly ordered system, proving that even in the world of abstract numbers, there is always a pattern waiting to be found.

Technical Summary: Additive Bases from Primitive Dyck Words

Problem Statement
This paper investigates additive representations of positive even integers using summands drawn from the set $N_pD$ . This set consists of zero and the integer values of primitive Dyck words interpreted as binary numbers. A Dyck word is a balanced sequence of 1s (up-steps) and 0s (down-steps); a word is primitive if its associated path returns to height zero only at the very end. The central question is to determine the additive order of this set: specifically, the least integer $h$ such that every sufficiently large even integer can be expressed as a sum of at most $h$ elements from $N_pD$ , and to classify the exact set of "exceptional" integers that require more than a given number of summands.

Methodology
The authors employ a hybrid approach combining formal language theory, combinatorial number theory, and constructive algorithms:

Positional Block Coding: The core structural insight is a bijection between primitive Dyck words and two-coloured Motzkin words via a base-4 block coding. By pairing consecutive binary digits, the authors map primitive Dyck words (excluding the word "10") to base-4 words of the form $3w0$ , where $w$ is a two-coloured Motzkin word. This allows the authors to translate the non-regular Dyck language into a structure compatible with base-4 positional arithmetic.
Regular Underapproximation: To handle integers congruent to $0 \pmod 4$ , the authors construct an explicit regular underapproximation of the primitive Dyck language. This subset, defined by the regular expression $11(01|10)^*00$ , corresponds to Motzkin words composed solely of digits $\{1, 2\}$ . This regular subset is amenable to finite automata methods and provides a covering for multiples of 4.
Digit Lifting and Propagation: The paper introduces an Interval Digit-Lifting Theorem. This theorem proves that if a set $A$ satisfies a closure property (specifically $4A + \{1, 2\} \subseteq A$ ), then a finite interval of representable numbers can be recursively extended to an infinite tail. This allows the authors to convert finite computational certificates (sumsets of small numbers) into proofs of representability for all sufficiently large integers.
Finite Certificates and Exhaustive Search: The proof relies on exact integer arithmetic to compute sumsets of small generators. The authors verify specific intervals where the sumsets of the regular family (for multiples of 4) and the full Motzkin-coded family (for integers $\equiv 2 \pmod 4$ ) overlap sufficiently to trigger the digit-lifting propagation.

Key Contributions

Exact Base-4 Characterization: The paper establishes the identity $DP = \{10\} \cup \phi(3 \cdot \text{Motz} \cdot 0)$ , where $\phi$ is a uniform morphism mapping base-4 digits to binary pairs. This provides a precise structural link between primitive Dyck words and Motzkin paths that preserves canonical positional values.
Classification of Exceptional Integers: The authors completely classify the positive even integers requiring more than six summands.
- The integer 46 is the unique even integer requiring eight summands.
- The integers 34, 44, 98, 154, 198, 202, 206, 838, 842, and 846 require exactly seven summands.
- Every other positive even integer requires at most six.
Sharp Thresholds: The paper identifies 848 as the sharp eventual threshold: every even integer $N \ge 848$ is a sum of at most six primitive Dyck numbers.
Asymptotic Order Analysis: The authors distinguish between the "relative order" (for all even integers) and the "asymptotic order" (for sufficiently large integers).
- The set $N_pD$ is a relative additive basis of order 8 for the even integers (due to the exception at 46).
- The relative asymptotic order is 6.
- The "halved" family $P$ (where $N_pD = \{0, 2\} \cup 4P$ ) has an asymptotic additive order of 5.
Constructive Algorithms: The digit-lifting mechanism yields a constructive algorithm that generates representations for large integers in $O(\log n)$ recursive stages, converting finite seed representations into infinite tails.

Results and Significance
The paper demonstrates that while the language of primitive Dyck words is deterministic context-free (and thus not regular), its additive properties can be effectively analyzed by separating the problem into residue classes modulo 4.

For $N \equiv 0 \pmod 4$ , a regular underapproximation suffices to prove that six summands are eventually sufficient.
For $N \equiv 2 \pmod 4$ , the proof requires the full power of the Motzkin-coded family and a five-summand covering, which, combined with the necessity of using an odd number of the summand '2', raises the asymptotic bound to six.

The significance of the work lies in its successful application of regular underapproximations to a non-regular language to solve an additive number theory problem. It provides a concrete example where the interplay between formal language structure (Dyck vs. Motzkin) and positional numeration systems yields sharp, computable bounds on additive bases. The authors explicitly note that the bound of six is asymptotically optimal, as the sequence $N_k = 10 \cdot 4^{k+1} - 6$ requires exactly six summands for every $k \ge 2$ .

All finite certificates and the classification of exceptional values are verified by supplementary programs using exact integer arithmetic, ensuring the reproducibility of the results. The paper concludes that 848 is the least even integer such that all larger even integers are representable by at most six primitive Dyck numbers, while 46 remains the unique exception requiring eight.

The Story of the Binary Balancers

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