Minkowski–Hlawka theorem

Remarks

Minkowski stated the result, and Edmund Hlawka gave the first proof in 1943. A standard proof averages over lattices. Siegel's mean value theorem says that for a random lattice $\Lambda$ of covolume $1$ and an integrable $f$, the mean of $\sum_{v \in \Lambda \setminus \set{0}} f(v)$ is $\int_{\R^n} f$. Applied to the indicator of a ball of volume $V < 1$, this gives a lattice of covolume $1$ with no nonzero point in the ball. Balls of half the radius centred at its points then form a packing of density $V/2^n$. Counting only primitive vectors and using the symmetry $v \mapsto -v$ improves the condition to $V < 2\zeta(n)$, which gives the bound $\zeta(n)/2^{n-1}$. For $n = 2$ the bound is $\zeta(2)/2 = \pi^2/12 \approx 0.82$, below the true optimum $\pi/\sqrt{12} \approx 0.91$. The argument is not constructive: it shows that good lattices exist without exhibiting one.

Going deeper

For a student

The lower bound $\zeta(n) 2^{1-n}$ is roughly $2^{-n}$, and upper bounds such as Rogers' bound and the Kabatiansky–Levenshtein bound are also exponentially small, but with a different exponent. Closing this exponential gap is a central open problem. Later work improved the lower bound by factors linear in $n$, for example Ball's bound $2(n - 1) \zeta(n) 2^{-n}$, and Venkatesh obtained a further $\log \log n$ factor in infinitely many dimensions. The same averaging idea underlies existence proofs for good error-correcting codes.