Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms

Abstract

We prove that if $\phi: {\Bbb R}^d \times {\Bbb R}^d \to {\Bbb R}$, $d \ge 2$, is a homogeneous function, smooth away from the origin and having non-zero Monge-Ampere determinant away from the origin, then $$ R^{-d} # \{(n,m) \in {\Bbb Z}^d \times {\Bbb Z}^d: |n|, |m| \leq CR; R \leq \phi(n,m) \leq R+\delta \} \lesssim \max \{R^{d-2+\frac{2}{d+1}}, R^{d-1} \delta \}.$$ This is a variable coefficient version of a result proved by Lettington in \cite{L10}, extending a previous result by Andrews in \cite{A63}, showing that if $B \subset {\Bbb R}^d$, $d \ge 2$, is a symmetric convex body with a sufficiently smooth boundary and non-vanishing Gaussian curvature, then $$ # \{k \in {\mathbb Z}^d: dist(k, R \partial B) \leq \delta \} \lesssim \max \{R^{d-2+\frac{2}{d+1}}, R^{d-1} \delta \}. (*)$$ Furthermore, we shall see that the same argument yields a non-isotropic 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 ${\Bbb R}^d$.