Sharp weak-type estimates for maximal operators associated to rare bases

Let (Formula presented.) denote a nonempty translation invariant collection of intervals in (Formula presented.) (which we regard as a rare basis), and define the associated geometric maximal operator (Formula presented.) by (Formula presented.) We provide a sufficient condition on (Formula presented.) so that the estimate (Formula presented.) is sharp. As a corollary, we obtain sharp weak-type estimates for maximal operators associated to several classes of rare bases including unions of one-parameter infinite families of Córdoba, Soria, and Zygmund-type bases.