We consider an instance of the following problem: Parties P1Pk each receive an input xi, and a coordinator (distinct from each of these parties) wishes to compute f(x1xk) 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 they 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