Hardy-Littlewood system and representations of integers by an invariant polynomial

Abstract

Let f be an integral homogeneous polynomial of degree d, and let $V_m = {X : f(X) = m}$ be the level set for each $m \in \mathbb{N}$. For a compact subset Ω in $V_1(\mathbb{R})$), set

$$ N_m(f, \Omega)=\# V_m(\mathbb{Z})\cap m^{1/d}\Omega\,. $$

We define the notion of Hardy-Littlewood system for the sequence {V _m }, according as the asymptotic of $N_m(f, \Omega)$as $n \rightarrow \infty$ coincides with the one predicted by Hardy-Littlewood circle method. Using a recent work of Eskin and Oh [EO], we then show for a large family of invariant polynomialsf, the level sets {V _m } are Hardy-Littlewood. In particular, our results yield a new proof of Siegel mass formula for quadratic forms.