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Long shortest vectors in low dimensional lattices

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 $γ_3$. We obtain a similar result for the generalisation of our construction to $4$ and $5$ dimensions.