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STNB2017, (31st edition)

Lattice points in elliptic paraboloids

Presenters

Carlos Pastor

Abstract

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 d2 for a wide family of truncated elliptic paraboloids in at least three dimensions, and certainly not smaller than d2 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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