Abstract
Let β be a real number greater than one, and let ℤβ be the set of real numbers which have a zero fractional part when expanded in base β. We prove that β is a Pisot number when the set ℕβ−ℕβ−ℕβ is discrete, where ℕβ=ℤβ∩[0,∞[. We also give partial answers to some related open problems, and in particular, we show that β is a Pisot number when a sum ℤβ+⋯+ℤβ is a Meyer set.