Lebesgue integral variant resolves the infinite word challenge in combinatorics, indicating a breakthrough in additive sequences.
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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Ingrid Vukusic (2025) studied this question.
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