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Counterexamples to a conjecture of Woods. (English) Zbl 1380.11086

Well rounded lattices are those, whose minimal vectors span the euclidean space. A. C. Woods conjectured in [J. Number Theory 4, 157–180 (1972; Zbl 0232.10020)] that the standard lattice \({\mathbb Z} ^n\) realises the maximum covering radius among all well rounded \(n\)-dimensional lattices of covolume 1. The present paper shows that this is not the case for all \(n\geq 30\) by explicitly constructing a series of lattices \(L_n = \alpha _1 \Lambda _{15} \perp \alpha _2 {\mathbb Z} ^{n-{15}}\) having larger covering radius than \({\mathbb Z}^n\). Here \(\Lambda _{15}\) is the laminated lattice of dimension 15 and \(\alpha _1,\alpha _2 \in {\mathbb R}_{>0}\) are uniquely determined by the property that \(L_n\) is well rounded of covolume 1.

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 0232.10020
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