We study the problem of authenticated multi-dimensional range queries over outsourced databases, where an owner outsources its database to an untrusted server, which maintains it and answers queries to clients. Previous schemes either scale exponentially in the number of query dimensions, or rely on heuristic data structures without provable bounds. Most importantly, existing work requires an exponential, in the database attributes, number of structures to support queries on every possible combination of dimensions in the database. In this paper, we propose the first schemes that (i) scale linearly with the number of dimensions, and (ii) support queries on any set of dimensions with linear in the number of attributes setup cost and storage. We achieve this through an elaborate fusion of novel and existing set-operation sub-protocols. We prove the security of our solutions relying on the q-Strong Bilinear Diffie-Hellman assumption, and experimentally confirm their feasibility.