One-round multi-party communication complexity of distinguishing sums

We consider an instance of the following problem: Parties P1, . . . , Pk each receive an input xi, and a coordinator (distinct from each of these parties) wishes to compute f(x1, . . . , xk) for some predicate f . We are interested in one-round protocols where each party sends a single message to the coordinator; there is no communication between the parties themselves. What is the minimum communication complexity needed to compute f , possibly with bounded error? We prove tight bounds on the one-round communication complexity when f corresponds to the promise problem of distinguishing sums (namely, determining which of two possible values the {xi} sum to) or the problem of determining whether the {xi} sum to a particular value. Similar problems were studied previously by Nisan and in concurrent work by Viola. Our proofs rely on basic theorems from additive combinatorics, but are otherwise elementary.