A characterization of piecewise $$\mathscr {F}$$ -syndetic sets

Some filter relative notions of size, \(\left( \mathscr {F},\mathscr {G}\right) \)-syndeticity and piecewise \(\mathscr {F} \)-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk (Semigroup Forum 79: 531–539, 2009). These notions are generalizations of the well studied notions of syndeticity and piecewise syndeticity. Since then, there has been an effort to develop the theory around the algebraic structure of the Stone–Čech compactification so that it encompasses these new generalizations. We prove one direction of a characterization of piecewise \(\mathscr {F}\)-syndetic sets. This completes the characterization, as the other direction was proved by Christopherson and Johnson (Semigroup Forum 104: 28–44, 2021).