Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers S is called invariant if it is closed under addition and multiplication, namely, for any x, y ∈ S we have x+y ∈ S and xy ∈ S . For each s ∈ C the smallest invariant set N[s] containing s consists of all possible sums ∑ i ∈ I aᵢ sⁱ , where I runs over all finite nonempty subsets of the set of positive integers N and aᵢ ∈ N for each i ∈ I . In this paper, we prove that for s ∈ C the set N[s] is everywhere dense in C if and only if s ∉ R and s is not a quadratic algebraic integer. More precisely, we show that if s ∈ C R is a transcendental number, then there is a positive integer n such that the sumset N tⁿ+N t²ⁿ +N t³ⁿ is everywhere dense in C for either $t=s$ or t=s+s² . Similarly, if s ∈ C R is an algebraic number of degree d ≠ 2, 4 , then there are positive integers $n, m$ such that the sumset N tⁿ+N t²ⁿ +N t³ⁿ is everywhere dense in C for t=ms+s² . For quadratic and some special quartic algebraic numbers s it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of N[s] in C is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.