On the normality of the concatenated Fibonacci constant

Abstract

We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of $\mathcal{F}$ in base $10$ would follow if almost all Fibonacci numbers were $(\varepsilon,k)$-normal in base $10$. The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first $500{,}000$ Fibonacci numbers in bases $10$ and $2$ indicate that global single-digit counts and $k$-block statistics for $k = 2, 3, 4$ are compatible with iid-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of $F_{n}$.