A Lebesgue variant of the additive square problem

TL;DR

This paper addresses a Lebesgue integral variant of the additive square problem in combinatorics on words, using Lebesgue's density theorem to provide a solution.

Contribution

It introduces a novel Lebesgue integral approach to a classic combinatorial problem, expanding the methods used in the field.

Findings

Solved the Lebesgue integral variant of the additive square problem

Applied Lebesgue's density theorem to combinatorial word problems

Provided new insights into the structure of infinite words with sum constraints

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.