The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k

Abstract

An infinite word x over an alphabet with b letters has full complexity if for each m 2 N all the b m words of length m are factors of x. We prove that the periods of˙1=3 k in the 2-adic expansion approach full complexity for increasing k: For any m 2 N, the periods for k > d.m C 1/ ln.2/= ln.3/e 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 .

The repetends of reduced fractions $a/b^k$ approach full complexity for increasing $k$

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