Variations on a Theme of Cassels for Additive Bases

We introduce the notion of caliber, cal (A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B. We further consider the limit superior t(A) of the average order of the number of representations of an integer as a sum of two elements of A. We give some basic properties of each notion and we relate the two together, thus yielding a generalization, of the form t(A) ≤ t(B)/ cal (A, B), of a result of Cassels specific to the case where A is an additive basis of the natural numbers and B is the sequence of perfect squares. We also provide some formulas for the computation of t(A) in a large class of cases, and give some examples.