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doi:10.1016/0304-3975(95)00234-0 
Copyright © 1996 Published by Elsevier Science B.V.
Regular paper
On the vector space of the automatic reals
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Communicated by M. Nivat
Abstract
A sequence (an)n 
0 is said to be k-automatic if an is a finite-state function of the base-k digits of n. We say a real number is (k, b)-automatic if its fractional part has a base-b expansion that forms a k-automatic sequence, and we denote the set of all such numbers as L(k,b). Lehr (Theoret. Comput. Sci. 108 (1993) 385–391) proved that L(k, b) forms a vector space over Q. In this paper we give a shortened version of the proof of Lehr's result and, answering a question of Bach, show that the dimension of the vector space L(k, b) is infinite.
We also give an example of a transcendental number such that all of its positive powers are automatic. The proof requires examining the coefficient of Xn in the formal power series (X + X2 + X4 + X8 + …)r. Along the way we are led to examine several sequences of independent combinatorial interest.
Finally, solving an open problem, we show that the automatic reals are not closed under (1) product; (2) squaring; and (3) reciprocal.
