A survey and a result on inhomogeneous quadratic forms

Abstract

We survey recent work done on the values at integer points of irrational inhomogeneous quadratic forms, namely, inhomogeneous analogues of the famous Oppenheim conjecture. We also prove that the set of such forms in two variables whose set of values at integer points avoids a given countable set not containing zero, has full Hausdorff dimension. Moreover, we consider the more refined variant of this problem, where the shift is fixed and the form is allowed to vary. The strategy is to translate the problems to homogeneous dynamics and deduce the theorems from their dynamical counterparts. While our approach is inspired by the work of Kleinbock and Weiss, the dynamical results can be deduced from more general results of An, Guan, and Kleinbock.