A survey and a result on inhomogeneous quadratic forms

TL;DR

This paper surveys recent advances on the values of irrational inhomogeneous quadratic forms at integer points and proves that certain sets of such forms have full Hausdorff dimension, using dynamical systems methods.

Contribution

It provides a survey of recent work and establishes new results on the Hausdorff dimension of inhomogeneous quadratic forms avoiding specific sets, extending the understanding of their distribution.

Findings

Set of inhomogeneous quadratic forms avoiding a countable set has full Hausdorff dimension.

Dynamical systems approach effectively analyzes the distribution of quadratic form values.

Results generalize previous work by Kleinbock, Weiss, and others.

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.