Abstract
We prove that a nonreal algebraic number theta with modulus greater than 1 is a complex Pisot number if and only if there is a nonzero complex number lambda such that the sequence of fractional parts ({R(lambda theta(n))})(n is an element of N) has a finite number of limit points. Also, we characterise those complex Pisot numbers theta for which there is a convergent sequence of the form ({R(lambda theta(n))})(n is an element of N) for some lambda is an element of C*. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.