\input zb-basic
\input zb-ioport
\iteman{io-port 06110229}
\itemau{L\'opez, Josefina; Stoll, Peter}
\itemti{The 2-adic, binary and decimal periods of $1/3^k$ approach full complexity for increasing $k$.}
\itemso{Integers 12, No. 5, 907-928 (2012).}
\itemab
Summary: An infinite word x over an alphabet with b letters has full complexity if for each $m\in \Bbb N$ all the $b^{m}$ words of length m are factors of x. We prove that the periods of $\pm 1/3^k$ in the 2-adic expansion approach full complexity for increasing k: For any $m\in \Bbb N$, the periods for $k>\lceil (m+1)\ln (2)/\ln (3)\rceil$ have complexity $2^m$. Amazingly, these $2^m$ words occur in the period almost the same number of times. On the way, first we prove the same for the binary period. We get a similar result for the decimal period of $1/3 ^{ k}$.
\itemrv{~}
\itemcc{}
\itemut{period of $1/3^k$; 2-adic; binary; decimal; complexity function of a word; frequency of factors; periodicity conjecture; $3x+1$}
\itemli{doi:10.1515/integers-2012-0013}
\end