  We begin by proving that any Presburger-definable image of one or more sets
of powers has zero natural density. Then, by adapting the proof of a dichotomy
result on o-minimal structures by Friedman and Miller, we produce a similar
dichotomy for expansions of Presburger arithmetic on the integers. Combining
these two results, we obtain that the expansion of the ordered group of
integers by any number of sets of powers does not define multiplication.
