Representations of Integers by an Invariant Polynomial and Unipotent Flows

Let f be an integral homogeneous polynomial of degree d in n variables. A basic problem in Diophantine analytic number theory is to understand the behavior of the integral representations of integers m by f as m tends to infinity. For each m ∈ N, consider the level variety Vm := {x ∈ Rn : f(x) = m}. For instance, if f(x1, · · · , xn) = x1 + · · · + xn (n ≥ 3), then Vm is the sphere of radius √ m centered at the origin and the set Vm(Z) = Vm ∩ Zn consists of integral vectors the sum of whose squares is equal to m. In this case, the asymptotic of #Vm(Z) is well known. For n ≥ 5 the classical Hardy-Littlewood circle method applies and for n = 4 the Kloosterman sum method works. For n = 3, Linnik gave a conditional answer and later Iwaniec gave a complete answer (see [Sa], [Iw]). In the case when Vm is non-compact, the number #Vm(Z) may be infinite. In this case one asks if there exists an asymptotic density for Vm(Z) as m → ∞ . To be more precise, for a compact subset Ω of V1, set