Long shortest vectors in low dimensional lattices
Abstract
For coprime integers $N,a,b,c$, with $0<a<b<c<N$, we define the set $$ \{ (na \! \! \! \! \pmod{N}, nb \! \! \! \! \pmod{N}, nc \! \! \! \! \pmod{N}) : 0 \leq n < N\}. $$ We study which parameters $N,a,b,c$ generate point sets with long shortest distances between the points of the set in dependence of $N$ and relate such sets to lattices of a particular form. As a main result, we present an infinite family of such lattices with the property that the normalised norm of the shortest vector of each lattice converges to the square root of the Hermite constant $\gamma_3$. We obtain a similar result for the generalisation of our construction to $4$ and $5$ dimensions.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2006.00461
 arXiv:
 arXiv:2006.00461
 Bibcode:
 2020arXiv200600461P
 Keywords:

 Mathematics  Number Theory;
 11P21;
 52C07
 EPrint:
 12 pages, 4 Figures. Generalisation to dimensions 4 and 5 added