On additive and multiplicative Hilbert cubes

Given subset E of natural numbers FS ( E ) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS ( E ) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε > 0 there exists a sequence with density > 1 − ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.