Lattice points close to families of surfaces, nonisotropic dilations and regularity of generalized Radon transforms

We prove that if φ : R × R → R, d ≥ 2, is a homogeneous function, smooth away from the origin and having nonzero Monge–Ampere determinant away from the origin, then R#{(n,m) ∈ Z × Z : |n|, |m| ≤ CR;R ≤ φ(n,m) ≤ R+ δ} . max{R 2 d+1 , Rδ}. This is a variable coefficient version of a result proved by Lettington, 2010, extending a previous result by Andrews, 1963, showing that if B ⊂ R, d ≥ 2, is a symmetric convex body with a sufficiently smooth boundary and nonvanishing Gaussian curvature, then (∗) #{k ∈ Z : dist(k,R∂B) ≤ δ} . max{R 2 d+1 , Rδ}. Furthermore, we shall see that the same argument yields a nonisotropic analog of (∗), one for which the exponent on the right hand side is, in general, sharp, even in the infinitely smooth case. This sheds some light on the nature of the exponents and their connection with the conjecture due to Wolfgang Schmidt on the distribution of lattice points on dilates of smooth convex surfaces in R.