Finite sums and idempotency for dynamically central piecewise syndetic sets

A set of positive integers is a set of pointwise recurrence if it supplies a return of every point to every neighborhood of itself in every minimal topological dynamical system. We prove that every such set contains finite-sums configurations of every finite rank. More generally, this holds after intersection with a return set to a neighborhood of a point in a minimal system: every dynamically central piecewise syndetic set is IP₀. The finite-sums configurations can be chosen with a piecewise syndetic set of simultaneous completions; their generators may be taken superincreasing.

An elementary consequence of established central-shift structure also shows that every upward closed, partition regular family F lying between the central sets C and the piecewise syndetic sets satisfies F ⊆ F + C ⊆ F + F. These results answer two questions posed by Glasscock and Le in 2024. The proofs combine their piecewise syndeticity theorem with elementary compactness and a partition argument using central shifts.

GitHub repository: eulogica/ramsey-finite-sums-idempotency