Representations of a number in an arbitrary base with unbounded digits

Abstract

In this paper, we prove that, for β ∈ ℂ {\beta\in{\mathbb{C}}} , every α ∈ ℂ {\alpha\in{\mathbb{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 {\alpha=d_{n}\beta^{n}+d_{n-1}\beta^{n-1}+\dots+d_{0}} with nonnegative integers n , d n , d n - 1 , … , d 0 {n,d_{n},d_{n-1},\dots,d_{0}} if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ {{\mathbb{Q}}} (possibly β itself) in the real interval ( 1 , ∞ ) {(1,\infty)} . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ ( 1 , ∞ ) {{\mathbb{C}}\setminus(1,\infty)} , there is α ∈ ℚ ⁢ ( β ) {\alpha\in{\mathbb{Q}}(\beta)} with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka.