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 $\Lambda<\mathbb{R}^d$, and if we set $k=\max\{j:\lambda_j\le1\}$, then $K$ contains at most $2^k(1+\frac{\lambda_k}2)^k/\lambda_1\cdots\lambda_k$ lattice points. This provides improved bounds in a conjecture of Betke, Henk and Wills (1993), and verifies that conjecture asymptotically as $\lambda_k\to0$. We also obtain a similar result without the symmetry assumption.