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Automata, algebraicity and distribution of sequences of powers

Jean-Paul Allouche, Jean-Marc Deshouillers, Teturo Kamae, Tadahiro Koyanagi (2001)

Annales de l’institut Fourier

Let K be a finite field of characteristic p . Let K ( ( x ) ) be the field of formal Laurent series f ( x ) in x with coefficients in K . That is, f ( x ) = n = n 0 f n x n with n 0 𝐙 and f n K ( n = n 0 , n 0 + 1 , ) . We discuss the distribution of ( { f m } ) m = 0 , 1 , 2 , for f K ( ( x ) ) , where { f } : = n = 0 f n x n K [ [ x ] ] denotes the nonnegative part of f K ( ( x ) ) . This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic distribution for f with f n 0 for some n < 0 . This distribution is...

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