On the Exceptional Sets of Integral Quadratic Forms

Abstract

A collection \mathcal S of equivalence classes of positive definite integral quadratic forms in n variables is called an n-exceptional set if there exists a positive definite integral quadratic form, which represents all equivalence classes of positive definite integral quadratic forms in n variables except those in S. We show that, among other results, for any given positive integers m and n, there is always an n-exceptional set of size m and there are only finitely many of them.