| Abstract [eng] |
A set of complex numbers is called invariant if it is closed under addition and multiplication, namely, for any we have and. For each the smallest invariant set containing consists of all possible sums, where runs over all finite nonempty subsets of the set of positive integers and for each. In this paper, we prove that for the set is everywhere dense in if and only if and is not a quadratic algebraic integer. More precisely, we show that if is a transcendental number, then there is a positive integer such that the sumset is everywhere dense in for either or. Similarly, if is an algebraic number of degree, then there are positive integers such that the sumset is everywhere dense in for. For quadratic and some special quartic algebraic numbers it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of in is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets. |
