Several New Results On The Lattice Point Problem

We mainly studied three problems about the number of lattice points in a domain.The first one is the classical lattice point problem associated with a special class of finite type domains D (?)Rd(d?3):we give an upper and a lower bound of the error term generated by using the volume of the domain to estimate the number of lattice points.Studying the number of lattice points in a domain is a branch of analytic number theory and is related to many other mathematical fields such as harmonic analysis and spectral geometry.Hence,it has very important research significance.It can be traced back to the great mathematician Gauss's study on the lattice counting in the enlarged disk in the plane.Inspired by the work of Randol,Kr(?)tzel,Nowak and my supervisor Guo,we consider the number of lattice points in the domain D which is more general in some respects than the domain they considered.The second one is the optimal stretching problem associated with the domain D,and moreover,with the general finite type domains ? (?) Rd(d? 2).We apply a diagonal positive definite matrix with determinant 1 to the domain for stretching,then we give an asymptotic property of the matrices corresponding to the "optimal stretches" that make the stretched domains contain the most positive lattice points or the least nonnegative lattice points.We confirm that the "optimal domains" are asymptotic balanced.The optimal stretching problem is closely related to the "shape/eigenvalue optimal problem"in spectral geometry.These two issues have attracted a lot of interest in recent years.There are some work about the optimal stretching associated with domains with nonzeroGaussian curvature or with very flat boundary(such as triangle),and the fact is that the results of the two cases do not match and the results of the latter are mostly limited in planar case.Our work is to give result of the case where the doQain is in Rk(d?2)and the curvature is between "non-flat" and "flat" cases.The third one is the lattice counting associated with the shells D(r,t)with the "radius" r and the "thickness" t:assume that u is uniformly distributed in Td,then we prove that if r and t satisfy some quantitative assumptions,the variance of the number of lattice points in d(r,t)-u is asymptotic to its expectation,namely the volume of the shells D(r,t).Consider lattice points in thin annulus or higher dimensional shells derived from their physical background.The investigation of the spectrum of certain quantum systems leads to the problem.We extend the result of Colzani,Gariboldi,and Gigante on the shells generated by convex domains with everywhere positive Gaussian curvature to the shells generated by the special finite type domain D.In this paper we mainly use some methods in harmonic analysis.We use the knowledge of oscillatory integral to study the Fourier transform of the surface measure induced on the boundary of the domain,since the three problems above can be reduced to some series summations containing the Fourier transform of the indicator function of the domain.Hence,curvature plays an important role.The Fourier transform of the surface measure near the point where the Gaussian curvature vanish may have a poor decline in the direction of the normal at this point.At present,there are few results on such situations,especially in high dimensional spaces.The finite type domains we considered allow the points with zero Gaussian curvature,and this is the difficulty of our work and the difference from the work of others.