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- W1939341109 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 ${Bbb R}^d$." @default.
- W1939341109 created "2016-06-24" @default.
- W1939341109 creator A5071497714 @default.
- W1939341109 creator A5083342755 @default.
- W1939341109 date "2011-03-09" @default.
- W1939341109 modified "2023-10-12" @default.
- W1939341109 title "Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms" @default.
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