Counterexamples to a Conjecture of Woods

A conjecture of Woods from 1972 is disproved. A lattice in R is called well-rounded if its shortest nonzero vectors span R, is called unimodular if its covolume is equal to one, and the covering radius of a lattice Λ is the least r such that R = Λ + Br, where Br is the closed Euclidean ball of radius r. Let Nd denote the greatest value of the covering radius over all well-rounded unimodular lattices in R. In [Woo72], A. C. Woods conjectured that Nd = √ d/2, i.e., that the lattice Z realizes the largest covering radius among wellrounded unimodular lattices. Moreover, Woods proved this statement for d ≤ 6. In [McM05], McMullen proved that Woods’s conjecture implies a celebrated conjecture of Minkowski. Spurred by this result, Woods’s conjecture has been proved for d ≤ 9 by Hans-Gill, Kathuria, Raka, and Sehmi (see [KR14] and references therein), thus yielding Minkowski’s conjecture in those dimensions. In this note we prove: Theorem. There is c > 0 such that Nd > c d √ log d . For all d ≥ 30,