On the lattice discrepancy of ellipsoids of rotation

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.