Uniform upper bound for normalized Fibonacci MAP lengths

Prove that for every positive integer difference d, the maximum length A(d) of a monochromatic arithmetic progression in the Fibonacci word satisfies (A(d)−1)/d < √5/τ, where τ=(1+√5)/2.

Background

The paper establishes exact formulas for A(d) in terms of the rotation step distance g(d), and it shows that the values A(F_n) along Fibonacci differences have asymptotic ratio A(F_n)/F_n approaching √5/τ from above.

The authors explain that they were unable to improve their asymptotic bounds sufficiently to prove a uniform estimate for all differences. They therefore formulate the displayed inequality as a conjecture, with the subtraction of 1 accounting for the approach from above along Fibonacci differences.

References

Unfortunately, we were not able to realise the second point and so we leave this as a conjecture.

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Conjecture 4, subsection “The Pisano period, rank of apparition and factors of Fibonacci numbers”