Abstract :
Given a full sublattice Λ of the integer lattice Zn in Euclidean n-space, we ask the following question. What is the largest real number r such that there is a sublattice Λ′ of Λ′ oZn isometric to r−1 Λ? It is not difficult to show that r must be the square root of a rational number. The paper ‘Shrinking lattice Polyhedra’ by Cremona and Landau (1990) gives an upper bound for r, which Cremona and Landau prove is always attained when n ⩽ 4. In this paper I show that this bound is attained when n = 5, and give some counterexamples to show that it is not always attained when n = 6 or n ⩾ 8. My method uses the arithmetic and algebra of 2 × 2 matrices over the ring of Hurwitz quaternions.
