Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters

Shea D. Burns; Dennis Davenport; Shakuan Frankson; Conner Griffin; John H. Johnson Jr.; Malick Kebe

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arXiv · 2503.05579

Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters

Shea D. Burns·Dennis Davenport·Shakuan Frankson·Conner Griffin·John H. Johnson Jr.·Malick Kebe

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Abstract

We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-Čech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-Čch compactification, our particular use of derived sets adapt and generalize methods recently used by Griffin arXiv:2311.09436 to characterize relative piecewise syndetic sets. As an application, we define an algebraically interesting subset of the Stone-Čech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal.

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Shea D. Burns, Dennis Davenport, Shakuan Frankson, Conner Griffin, John H. Johnson Jr., Malick Kebe. 2025-12-15. Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters. https://arxiv.org/abs/2503.05579

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