Thue–Morse maximal odd-difference MAP lengths

Establish whether every odd difference d>1 in the Thue–Morse sequence satisfies A(d)\geq4, equivalently prove that no odd difference has longest MAP length A(d)=3.

Background

The paper discusses prior work showing that every odd difference d>1 in the Thue–Morse sequence has a monochromatic arithmetic progression of length at least three. It identifies the stronger lower bound four as an explicitly conjectured improvement, which the paper subsequently resolves.

Because the paper proves this conjecture in the following lemma, the passage records a previously open problem rather than an unresolved question in the final results of the paper.

References

The authors conjecture that this bound can be improved to A(d)\geq 4. That is, they claim that there exists no odd d for which A(d)=3.

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Section 3.2, immediately preceding Lemma 3.2