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In this paper, we prove a criterion for complexity in $g$-ary expansions of arational fraction $a/b<1$ with gcd$(a,b)=1$. We prove that for any purelyperiodic proper fraction $a/b$ and all $j\geq 1$, each sequence of $j$ digitsoccurs in the $g$-ary repetend of $a/b^k$ with a relative frequency thatapproaches $1/g^j$ with an increasing $k$. The absolute frequencies can becalculated by means of a simple transition matrix. Let $(a_k)$ be a sequence ofpositive integers relatively prime to $b$. We prove that each sequence of $j$digits occurs in the $g$-ary repetend of $a_k/b^k$ with a relative frequencythat approaches $1/g^j$ with an increasing $k$, unless all prime factors of $b$divide the base $g\geq 2$.

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