Automata, algebraicity and distribution of sequences of powers

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) = Sigma (infinity)(n=n0) f(n)x(n) with n(0) is an element of Z and f(n) is an element of K (n = n(0), n(0) + 1, (...)). We discuss the distribution of ({f(m)})m=0,1,2,(...) for f is an element of K((x)), where {f} := Sigma (infinity)(n=0) f(n)x(n) is an element of K[[x]] denotes the nonnegative part of f is an element of 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) not equal 0 for some n < 0. This distribution is not the uniform measure on K[[x]], but is equivalent to it. We have a different situation for f is an element of K[[x]], where if f(0) not equal 0 and f not equal f(0), then the distribution for f is continuous but has a small support. We prove in this case, that the distribution for f(-1) is identical with the distribution for f(0)(-2)f. Christol, Kamae, Mendes France and Rauzy proved that the algebraicity of f(x) is an element of K((x)) over K(x) is equivalent to the p-automaticity of the sequence (f(n)). This result was generalized to the multidimensional case by Salon. Hence, if the Laurent series f(x) is an element of K((x)) is algebraic over K(x), then F(x,y) := Sigma (infinity)(m=0) f(x)(m)y(m) is 2-dimensionally p-automatic, since it is algebraic over the field K(x, y). We construct a finite automaton recognizing the sequence of coefficients of this double series F(x,y) to discuss the distribution of ({f(m)})(m greater than or equal to0). Thus, we generalize results by Houndonougbo and Deshouillers, and strengthen results by Allouche and Deshouillers.