Uniform upper bound for Fibonacci-word MAP growth

Prove that every positive difference d in the Fibonacci word satisfies (A(d)-1)/d<\sqrt{5}\,\tau^{-1}.

Background

The authors derive exact formulas and asymptotic growth rates for several families of differences, including Fibonacci numbers and their multiples. They state that these results did not suffice to establish the desired global asymptotic bound on A(d).

The conjecture proposes a uniform inequality over all positive differences, with the subtraction of one accounting for the fact that the ratios A(F_n)/F_n approach the limiting slope from above.

References

Unfortunately, we were not able to realise the second point and so we leave this as a conjecture. \begin{conj}\label{CONJ:fibmaxslope} For all d \geq 1, \frac{A(d)-1}{d} < \frac{\sqrt{5}{\tau}.

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Section 4.4, subsection “The Pisano period, rank of apparition and factors of Fibonacci numbers,” immediately preceding Conjecture 4.12