# P2824: Infinite additive-square avoidance over a finite integer alphabet

- ID: `P2824`
- Reference: `additive-square-finite-alphabet`
- Page: https://theoremdb.org/statements/P2824
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a finite set \(A\subset\mathbb Z\) and an infinite word \(a_0a_1a_2\cdots\) with every \(a_i\in A\) such that \(\sum_{r=0}^{\ell-1}a_{i+r}\ne\sum_{r=0}^{\ell-1}a_{i+\ell+r}\) for every \(i\ge0\) and every \(\ell\ge1\)?

### Definitions

- **Definition.** An additive square is a pair of consecutive blocks of the same positive length and the same sum.
- **Definition.** Avoidance means that no choice of starting index \(i\) and block length \(\ell\) gives an additive square.
- **Definition.** The alphabet consists of integer values, so equality of block sums is part of the structure rather than a relabeling-invariant property.

### What counts as a solution

- Construct a finite integer alphabet and an infinite word with a proof of every displayed inequality, or prove that every infinite word over every finite integer alphabet contains an additive square.

## Status

The 2026-07-28 audit found neither an infinite additive-square-free word over a finite integer alphabet nor an impossibility proof; the exact finite maximum over \(\{0,1,2,4\}\) is 62, and the infinite existence question remains open. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (The finite-integer additive-square problem remains open).** The 2026-07-28 audit found neither an infinite additive-square-free word over a finite integer alphabet nor an impossibility proof; the exact finite maximum over \(\{0,1,2,4\}\) is 62, and the infinite existence question remains open.

Vukusic states the discrete question explicitly and calls it open, then proves a Lebesgue-integral analogue. Andrade and Mol, revised in February 2025, also state that the finite integer problem remains unknown and distinguish the known Z^2 construction. Popoli, Shallit, and Stipulanti give the same status in FSTTCS 2024. A dated search on 2026-07-28 checked these papers, the original formulations, exact-title queries, citation indexes, and nearby additive-power work. This record is a dated literature report and makes no claim about sources published after the search.

### Background and intake notes

This question links word avoidance with zero-sum intervals: the difference sequence between adjacent block sums must avoid zero at every scale. Finite extremal words, normalized alphabets, morphism tests, and unavoidable-pattern lemmas all form reusable partial results.

- Original intake status: UNKNOWN as of 2026-07-27. A 2025 primary source calls the existence of an infinite additive-square-free word over a finite alphabet open and settles a Lebesgue-integral analogue instead.
- 2026-07-27 status search checked the 2025 Lebesgue-variant paper, additive-power surveys, and recent additive word-complexity work; no discrete infinite construction or impossibility proof was found.
- Affine changes \(a\mapsto ua+v\) with nonzero integer \(u\) preserve additive-square avoidance, so searches may normalize the alphabet minimum and gcd without changing the target.
- Long finite words give lower bounds on the necessary obstruction length. A compact morphic, automaton, or entropy-compression certificate would carry more information than a lone record word.

- Recorded example: The finite word \(012\) contains no additive square: its adjacent one-letter blocks have different sums, and it is too short for two blocks of length two.

### Other known results

- **Computation 1** (reproduced): Every additive-square-free word over {0,1,2,4} has length at most 62. Exactly two length-62 words occur in the complete search tree, and they form one reversal orbit.
- **Claim 2** (supported): Every abelian square is an additive square under an integer labeling, and infinite abelian-square-free words require at least four letters. Therefore an additive-square-free infinite word needs an alphabet of size at least four. [4](#reference-4)
- **Claim 3** (supported): Freedman and Brown determine the longest good words for four-letter sets satisfying the endpoint-sum equation; the ten primitive normalized reflection classes with maximum letter at most 8 have maxima between 50 and 60. [5](#reference-5)
- **Claim 4** (supported): Rao and Rosenfeld prove that the fixed point of h6 has no adjacent equal-length blocks with equal vector sums under six explicit weights in Z^2. [3](#reference-3)

### Prior approaches

- **Route 1** (reproduced): An exact depth-first sweep closed all ten primitive normalized reflection representatives in the balanced family through maximum letter 8 and matched the published maxima.
- **Route 2** (reproduced): An exact screen of every primitive direction of coefficient height at most 1000 found 656,301 directions whose projected h6 word survives through 378,367 symbols; the smallest surviving direction is (383,37).
- **Route 3** (ruled out): The projected h6 candidate with direction (361,288) survives 200,000 symbols and then has its first scalar additive square ending at position 250,952.
- **Route 4** (ruled out): The Rao-Rosenfeld sufficient decision criterion cannot apply to a scalar projection of h6: its expanding eigenspace has dimension 3, while the length-and-sum map has rank at most 2. [3](#reference-3)
- **Route 5** (supported): The 2026-07-28 audit resolved the canonical target, found no attached research records or duplicate exact targets, and separated the open Z question from known Z^2, unequal-length, approximation, cube, and bounded-alphabet results. [1](#reference-1) [4](#reference-4) [2](#reference-2) [3](#reference-3) [5](#reference-5) [6](#reference-6) [7](#reference-7)

### Open directions

- **Route 6** (reported): Synthesize a primitive morphism with at most two expanding eigendirections and a finite integer weighting, then apply the finite template-ancestor test under its valid spectral hypothesis. [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): This Python driver enumerates the primitive normalized balanced family through a chosen maximum letter and invokes the linked exact DFS on every reflection representative.
- **Artifact 2** (reproduced): This self-contained Python program exhausts the finite prefix tree for a supplied integer alphabet using exact prefix sums and checks every new suffix for adjacent equal-length equal-sum blocks.
- **Artifact 3** (reproduced): This independent C++20 implementation closes the full {0,1,2,4} search tree and returns the same node count, leaf count, maximum length, and two reversal-related maximizers.
- **Artifact 4** (reproduced): This C++20 program converts each nonzero block-sum difference vector into its unique normalized perpendicular direction and marks the first scalar additive square for every bounded primitive projection.
- **Artifact 5** (reproduced): This C++20 program scans one primitive h6 projection in end-major order and returns the earliest scalar additive square with an exact interval and equal sums.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `additive-square-finite-alphabet`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Ingrid Vukusic, A Lebesgue variant of the additive square problem, arXiv:2506.21200v1, abstract and opening paragraph; cross-checked against Andrade and Mol, arXiv:2408.15390v2, Introduction, and Popoli, Shallit, and Stipulanti, LIPIcs.FSTTCS.2024.32, Introduction The source states the discrete additive-square problem and solves a continuous analogue; this CC0 self-contained restatement was prepared on 2026-07-27. https://arxiv.org/abs/2506.21200
   - Also cited at Ingrid Vukusic, A Lebesgue variant of the additive square problem, arXiv:2506.21200v1, abstract and opening paragraph; cross-checked against Andrade and Mol, arXiv:2408.15390v2, Introduction, and Popoli, Shallit, and Stipulanti, LIPIcs.FSTTCS.2024.32, Introduction
   - Also cited at abstract and opening paragraph
   - preprint; reference source; arXiv source revision v1; checked 2026-07-28
   - Source use: citation_only
   - States the discrete finite-alphabet additive-square question as open and proves a continuous analogue.
   - Source used to assess the problem's recorded status.
   - For Infinite additive-square avoidance over a finite integer alphabet: The 2026-07-28 audit resolved the canonical target, found no attached research records or duplicate exact targets, and separated the open Z question from known Z^2, unequal-length, approximation, cube, and bounded-alphabet results.
2. <a id="reference-2"></a>Introduction, additive-power discussion, Theorem 29 https://doi.org/10.4230/LIPIcs.FSTTCS.2024.32
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Studies additive powers with Walnut and gives a current source for the open finite-integer questions.
3. <a id="reference-3"></a>Rao and Rosenfeld, arXiv:1511.05875v2, h6 Jordan decomposition in the Results section and the E_e(M_h) intersect ker(Phi) criterion in the Applications section; rank-nullity deduction checked 2026-07-28 Michael Rao and Matthieu Rosenfeld, Avoiding two consecutive blocks of same size and same sum over Z^2, arXiv:1511.05875v2, Results section, displayed h6 morphism and subsection Additive-square-free words on Z^2 https://arxiv.org/abs/1511.05875
   - Also cited at Applications and Results sections
   - Also cited at Rao and Rosenfeld, arXiv:1511.05875v2, h6 Jordan decomposition in the Results section and the E_e(M_h) intersect ker(Phi) criterion in the Applications section; rank-nullity deduction checked 2026-07-28
   - Also cited at Rao and Rosenfeld, arXiv:1511.05875v2, Applications section, proposition on finite parents and theorem deciding k-th-power-modulo-Phi freeness
   - preprint; reference source; arXiv:1511.05875v2; checked 2026-07-28
   - Source use: citation_only
   - Constructs an additive-square-free morphic word over six weights in Z², a neighboring result with a different alphabet space.
4. <a id="reference-4"></a>Jonathan Andrade and Lucas Mol, Avoiding abelian and additive powers in rich words, arXiv:2408.15390v2, Introduction, definitions and paragraphs immediately preceding the additive-power status discussion https://arxiv.org/abs/2408.15390
   - Also cited at Introduction and decision-algorithm section
   - preprint; reference source; arXiv:2408.15390v2; checked 2026-07-28
   - Source use: citation_only
   - Records the current open status of the exceptional additive-cube class and the finite-integer additive-square problem.
5. <a id="reference-5"></a>Allen R. Freedman and Thomas C. Brown, Sequences on Sets of Four Numbers, Integers 16 (2016), A33, Theorem 1 and the table on pages 2-3. Allen R. Freedman and Thomas C. Brown, Sequences on Sets of Four Numbers, Integers 16 (2016), A33, Theorem 1 and the table on pages 2-3 https://math.colgate.edu/~integers/q33/q33.pdf
   - Also cited at Theorem 1 and pages 2-3
   - website; reference source; PDF checked 2026-08-01; checked 2026-07-28
   - Source use: citation_only
   - Determines exact finite maxima for a normalized family of four-letter integer alphabets.
6. <a id="reference-6"></a>Approximations of additive squares in infinite words Theorem 2.1 https://doi.org/10.1515/integers-2012-0006
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Proves an approximation theorem for additive squares in infinite words without resolving exact avoidance.
7. <a id="reference-7"></a>An application of van der Waerden's theorem in additive number theory main theorem and question on final page https://math.colgate.edu/~integers/a7/a7.pdf
   - website; reference source; PDF checked 2026-08-01; checked 2026-07-28
   - Source use: citation_only
   - Provides an earlier finite-word result used to compare exact additive-square maxima.