Carlos Pastor
Given a compact subset K of Rd, let N(R) denote the number of points in the lattice Zd that lie within K after being dilated by a factor R>1. The lattice point problem associated to K consists in approximating N(R) for large R, usually in the form N(R)=|K|Rd+O(Rα) where |K| stands for the volume of K. Determining how small the exponent α can be taken in this asymptotic formula is an open problem even for simple regions such as the circle (Gauss problem) or rational ellipsoids in the three-dimensional space. In this talk I will present a recent joint work with F. Chamizo, showing that α may be taken arbitrarily close to d−2 for a wide family of truncated elliptic paraboloids in at least three dimensions, and certainly not smaller than d−2 in some cases. To do this the problem is first reduced to the estimation of a 2-dimensional quadratic exponential sum, which is then bounded using a simplified version of the circle method.
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