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

TL;DR

This paper generalizes Khintchine-type theorems to subhomogeneous functions, establishing conditions for their values at integer points to be well-approximated, extending previous results on quadratic forms and polynomials.

Contribution

It introduces a broad framework for approximation results of subhomogeneous functions under group actions, using Rogers' second moment estimates.

Findings

Derived necessary and sufficient conditions for $ ext{psi}$-approximability of $f \

Established a criterion for uniform approximation of subhomogeneous functions.

Extended Khintchine-type theorems to a general class of functions and group actions.

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 $\psi$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $\psi$-approximable; that is, $|f\circ g(\mathbf{v})| \le \psi(\|\mathbf{v}\|)$ 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.