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

Conner Griffin

Published 2023-11-15Version 1

Some filter relative notions of size, $\left( \mathcal{F},\mathcal{G}\right) $-syndeticity and piecewise $\mathcal{F} $-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk in their paper ``The closure of the smallest ideal of an ultrafilter semigroup.'' 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-\v{C}ech compactification so that it encompasses these new generalizations. In this direction, we prove a characterization of piecewise $\mathcal{F}$-syndetic sets. This fully answers a conjecture of Christopherson and Johnson. arXiv:2105.09723