  We consider a potential outcomes model in which interference may be present
between any two units but the extent of interference diminishes with spatial
distance. The causal estimand is the global average treatment effect, which
compares outcomes under the counterfactuals that all or no units are treated.
We study a class of designs in which space is partitioned into clusters that
are randomized into treatment and control. For each design, we estimate the
treatment effect using a Horvitz-Thompson estimator that compares the average
outcomes of units with all or no neighbors treated, where the neighborhood
radius is of the same order as the cluster size dictated by the design. We
derive the estimator's rate of convergence as a function of the design and
degree of interference and use this to obtain estimator-design pairs that
achieve near-optimal rates of convergence under relatively minimal assumptions
on interference. We prove that the estimators are asymptotically normal and
provide a variance estimator. For practical implementation of the designs, we
suggest partitioning space using clustering algorithms.
