Khintchine-type theorems for values of subhomogeneous functions at integer points

Abstract

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $f: \mathbb{R}^n \to \mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $ψ$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $ψ$-approximable; that is, $|f\circ g(\mathbf{v})| \le ψ(\|\mathbf{v}\|)$ for infinitely many $\mathbf{v} \in \mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $\rm{ASL}_n(\mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates.