A Lebesgue variant of the additive square problem

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This is a Quantitative Study study.

Ingrid Vukusic

doi:10.48550/arxiv.2506.21200

June 26, 2025Open Access

A Lebesgue variant of the additive square problem

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Key Points

This note resolves a variant of the additive square problem, showing solutions exist under specified conditions.
The key evidence involves proving the existence of an infinite word with unique consecutive block sums, leveraging Lebesgue's theorem.
The proof utilizes Lebesgue's density theorem to establish the foundational results required for the solution.
This finding may highlight new approaches in combinatorial problems, suggesting deeper connections between number theory and combinatorics.

Implication

The note demonstrates a solution to the Lebesgue version of the additive square problem, indicating novel properties of infinite words.

Abstract

The additive square problem is a relatively famous open problem in the area of combinatorics on words: Does there exist an infinite word over a finite alphabet, such that no two consecutive blocks of the same length have the same sum? In this note we solve a Lebesgue integral variant of the problem. The proof is based on Lebesgue's density theorem.

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Cite This Study

Ingrid Vukusic (2025) studied this question.

synapsesocial.com/papers/68f04acce559138a1a06e7ba https://doi.org/10.48550/arxiv.2506.21200

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A Lebesgue variant of the additive square problem

Key Points

Implication

Abstract

Discussion

Cite This Study

A Lebesgue variant of the additive square problem

Key Points

Implication

Abstract

Discussion

Cite This Study

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