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On the lattice discrepancy of ellipsoids of rotation

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Author(s): Werner Georg Nowak

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 4;
Issue: 2;
Start page: 101;
Date: 2009;
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Keywords: Lattice points | lattice discrepancy | convex bodies | bodies of rotation | ellipsoids | exponential sums.

ABSTRACT
The objective of this paper is to prove that the number of lattice points $A_\Epsilon_\alpha(t)$ in the elipsoid$${u_1^2+u_2^2\over a} + a^2 u_3^2 \le t^2 $$ satisfies the asymptotic$$ A_\Epsilon_\alpha(t) = {4\pi\over3}t^3 + \O{t^{679/494+\epsilon}}, $$ for fixed $a$, large $t$, and any $\epsilon > 0$. This improves upon the error term $\O{t^{11/8+\epsilon}}$ known before.

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