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We prove that if $\phi: {\Bbb R}^d \times {\Bbb R}^d \to {\Bbb R}$, $d \ge2$, is a homogeneous function, smooth away from the origin and having non-zeroMonge-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 thatif $B \subset {\Bbb R}^d$, $d \ge 2$, is a symmetric convex body with asufficiently 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-isotropicanalog of $(*)$, one for which the exponent on the right hand side is, ingeneral, sharp, even in the infinitely smooth case. This sheds some light onthe nature of the exponents and their connection with the conjecture due toWolfgang Schmidt on the distribution of lattice points on dilates of smoothconvex surfaces in ${\Bbb R}^d$.

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