Classification of finite maximal odd differences for Thue–Morse MAP lengths

Determine whether infinitely many integers n satisfy O_max(n)=-\infty, O_max(n)=\infty, or O_max(n)<\infty, and in particular decide whether the observed sets \{1,3\}, \{2,4,5\}, and the currently observed infinite-value cases exhaust the corresponding classifications.

Background

For each MAP length n, O_max(n) is defined as the largest odd difference d for which the Thue–Morse sequence has longest MAP length A(d)=n. The computed values show several different behaviors: some lengths have no odd preimage, some have a largest odd preimage, and some appear to have infinitely many odd preimages.

The authors report that computational limitations prevent them from calculating O_max(n) for n\geq8 and explicitly ask whether each of these observed behaviors persists infinitely often or whether the finite exceptional sets suggested by the data are complete.

References

Are there infinitely many values of n for which O_{\max}(n) = -\infty, or is {1,3} the full set of values for which O_{\max} = -\infty? Are there infinitely many values of n for which O_{\max}(n)=\infty? Are there infinitely many values of n for which O_{\max}(n)<\infty$, or is {2,4,5} the full set of values for which O_{\max} is finite?

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Section 3.2, subsection “Preimages of A(d),” Question 3.3