The famous Erdős distinct distances problem asks the following: how many distinct distances must exist between a set of n points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of g-rigidity. Specifically, if G=(V,E) is a (hyper)graph and g is a map assigning polynomial measurements to the (hyper)edges of G, our main results describe sharp lower bounds for the number of different edge-labellings that are achievable from realisations where all vertices are restricted to a point set P. This allows us to obtain results for pseudo-Euclidean metrics, Lp metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. This talk is presenting joint work with Nora Frankl, Samuel Mansfield, Anthony Nixon, Jonathan Passant and Audie Warren.