On sparse geometry of numbers (Q2226526)

Given a lattice \(L\) of full rank in \(n\)-dimensional real space, a vector in the lattice is called \(i\)-sparse if it has no more than \(i\) non-zero coordinates. Define the \(i\)-th successive sparsity level of \(L\), denoted \(s_i(L)\), to be the minimal \(s\) such that the lattice has \(s\) linearly independent \(i\)-sparse vectors. The authors give sufficient conditions for \(s_i (L)\) to be smaller than \(n\) and give explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in \(L.\) They use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. In the \(2\)-dimensional situation, they show that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real \(j\)-invariant. They identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.