2026/07/16 by Ralf Stephan
#math.CO #math.NT
We prove two results on the fine structure of the binary digits of 3m. First, for every fixed period p, the number of positions at which the binary expansion of 3m breaks p-periodicity grows in order like log m/loglog m; equivalently, no window of the expansion deeper than a fixed power of log m is p-periodic. Second, the finite binary word formed by the low-order digits of 3m has full low-order subword complexity: its complexity function satisfies \pcx3m(n)≥ n+1 for every length n, once m is large enough.