New bounds in the discrete analogue of Minkowski's second theorem

We adapt an argument of Tao and Vu to show that if ${\lambda }_1\le \cdots \le {\lambda }_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\mathrm{\Lambda }<{\mathbb{R}}^d$, and if we set $k=\max \{j:{\lambda }_j\le 1\}$, then $K$ contains at most $2^k(1+\frac{{\lambda }_k}{2}{)}^k\mathrm{/}{\lambda }_1\cdots {\lambda }_k$ lattice points. This provides improved bounds in a conjecture of Betke, Henk and Wills (1993), and verifies that conjecture asymptotically as ${\lambda }_k\to 0$. We also obtain a similar result without the symmetry assumption.