On the distribution of powers of a complex number

Abstract

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξαn + ν, n = 0, 1, 2, . . . , modulo \({\mathbb{Z}[i],}\) where \({i=\sqrt{-1}}\) and \({\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}}\) is the ring of Gaussian integers. For any \({z\in \mathbb{C},}\) one may naturally call the quantity z modulo \({\mathbb{Z}[i]}\)the fractional part of z and write {z} for this, in general, complex number lying in the unit square \({S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) 1 then there are two limit points of the sequence {ξαn +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \({\mathbb{Q}(i)}\) all lie in the unit disc |z| ≤ 1 and \({\xi\in\mathbb{Q}(\alpha,i).}\) Then we prove a result in the opposite direction which implies that, for any fixed \({\alpha\in\mathbb{C}}\) of modulus greater than 1 and any sequence \({z_n\in\mathbb{C},n=0,1,2,\dots,}\) there exists \({\xi \in \mathbb{C}}\) such that the numbers ξαn−zn, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \({\mathbb{Z}[i]}\). In particular, they all can be covered by a union of small discs with centers at \({(1+i)/2+\mathbb{Z}[i]}\) if |α| is large.