Thick Sets, Syndetic Sets, and a Lacunary Difference Set

Bounded gaps and arbitrarily long runs are complementary notions, and the complementarity is exact: a set of integers is thick precisely when its complement is not syndetic. We prove that duality, and then two facts that are easy to state backwards. A thick set does not have upper Banach density zero; it has density one, because it fills a window of every length completely. A syndetic set has positive lower Banach density, since it meets every window of its gap length. These are not two strengths of one notion: syndeticity bounds every window from below, while thickness fills one window per length. At the opposite extreme sits the lacunary set 2ⁿ, for which every non-zero difference pins both exponents, so no non-zero difference recurs and the difference set is $0$.