Representations of a number in an arbitrary base with unbounded digits

Abstract

In this paper, we prove that, for β ∈ ℂ , every α ∈ ℂ has at most finitely many (possibly none at all) representations of the form α = d n ⁢ β n + d n - 1 ⁢ β n - 1 + … + d 0 with nonnegative integers n , d n , d n - 1 , … , d 0 if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ (possibly β itself) in the real interval ( 1 , ∞ ) . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ ( 1 , ∞ ) , there is α ∈ ℚ ⁢ ( β ) with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka.