Small Sets Satisfying the Central Sets Theorem

The Central Sets Theorem is a powerful theorem, one of whose consequences is that any central set in N contains solutions to any partition regular system of homogeneous linear equations. Since at least one set in any finite partition of N must be central, any of the consequences of the Central Sets Theorem must be valid for any partition of N. It is a result of Beiglböck, Bergelson, Downarowicz, and Fish that if p is an idempotent in (βN,+) with the property that any member of p has positive Banach density, then any member of p satisfies the conclusion of the Central Sets Theorem. Since all central sets are members of such idempotents, the question naturally arises whether any set satisfying the conclusion of the Central Sets Theorem must have positive Banach density. We answer this question here in the negative.