Beta-expansion in Pisot and Salem unit bases in Fq((x −1 ))

Abstract

In [11] K. Schmidt studied the lengths of periods occurring in the β-expansion of a

rational number r noted by P erβ(r) for the Pisot numbers of a very special form satis-
fying β

2 = nβ+1 for some n ≥ 1. He showed the curious property "P erβ(
p
q
) = P erβ(
1
q
)"
for all positive integers p, q with p ∧ q = 1 and p < q. The aim of this paper is to prove
that in the case of a formal power series over finite fields this property is true for special
cubic Pisot unit basis.