β -expansion and transcendence in Fq((x−1))Fq((x−1))

The aim of this paper is to prove that the strings of 0 in the β-expansion of 1 exhibit a lacunarity bounded by a quantity related to the absolute value of the conjugate of β when β   is an algebraic power series over the finite field FqFq with |β|>1|β|>1. This result provides, in a natural way, a family of transcendental power series β   such that dβ(1)dβ(1) is not q-automatic.