 

Alex Iosevich and Krystal Taylor
Lattice points close to families of surfaces, nonisotropic dilations and regularity of generalized Radon transforms view print


Published: 
December 27, 2011 
Keywords: 
Lattice points, Harmonic analysis, Erdos problems. 
Subject: 
11P21, 42B35, 52C10 


Abstract
We prove that if ϕ: R^{d} × R^{d} → R, d ≧ 2, is a homogeneous function, smooth away from the origin and having nonzero MongeAmpere determinant away from the origin, then
R^{d} # {(n,m) ∈ Z^{d} × Z^{d}: n, m ≦ CR; R ≦ ϕ(n,m) ≦ R+δ }
\lesssim max{R^{d2+(d+1)/2}, R^{d1} δ}.
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}, d ≧ 2, is a symmetric convex body with a sufficiently smooth boundary and nonvanishing Gaussian curvature, then
(*) #{k ∈ Z^{d}: dist(k, R\partial B) ≦ δ } \lesssim
max{R^{d2+(d+1)/2}, R^{d1} δ}.
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^{d}.


Acknowledgements
This work was partially supported by the NSF Grant DMS1045404.


Author information
Alex Iosevich:
Department of Mathematics, University of Rochester, Rochester, NY
iosevich@math.rochester.edu
Krystal Taylor:
Department of Mathematocs, University of Rochester, Rochester, NY
taylor@math.rochester.edu

