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:R-n -> R, we derive a necessary and sufficient condition on the approximating function psi for guaranteeing that a generic element f circle g in the G-orbit of f is psi-approximable; that is, |f circle g(v)|<=psi(||v||) for infinitely many v is an element of Z(n). We also deduce a sufficient condition in the case of uniform approximation. Here G can be any closed subgroup of ASL(n)(R) satisfying certain axioms that allow for the use of Rogers-type estimates.