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

Dmitry Kleinbock, Mishel Skenderi

Published 2019-10-04Version 1

This work has been motivated by several recent papers quantifying the density of values of generic quadratic forms and other polynomials at integer points, in particular using Rogers second moment estimates (Athreya-Margulis, Kelmer-Yu). In this paper we exploit similar ideas in quite general set-up of arbitrary subhomogeneous functions, deriving necessary and sufficient conditions on approximating function $\psi$ guaranteeing that for generic $f$ in the $G$-orbit of a given function the inequality $|f(\mathbf{v})| \le \psi(\|\mathbf{v}\|)$ has finitely or infinitely many integer solutions. Here $G$ can be any group satisfying certain natural conditions guaranteeing that Rogers-type estimates can be applied.