On sparse geometry of numbers

Abstract

Let L be a lattice of full rank in n-dimensional real space. A vector in L is called i-sparse if it has no more than i nonzero coordinates. We define the ith successive sparsity level of L, \(s_i(L)\), to be the minimal s so that L has s linearly independent i-sparse vectors, then \(s_i(L) \le n\) for each \(1 \le i \le n\). We investigate sufficient conditions for \(s_i(L)\) to be smaller than n and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in L. These results can be viewed as a partial sparse analogue of Minkowski’s successive minima theorem. We then 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. We further investigate the 2-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real j-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.

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Acknowledgements

We wish to thank the anonymous referees for their helpful suggestions, which improved the quality of presentation.

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Correspondence to Lenny Fukshansky.

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Fukshansky was partially supported by the Simons Foundation Grant #519058

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Fukshansky, L., Guerzhoy, P. & Kühnlein, S. On sparse geometry of numbers. Res Math Sci 8, 2 (2021). https://doi.org/10.1007/s40687-020-00238-z

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Keywords

  • Lattices
  • Sparse vectors
  • Virtually rectangular lattices
  • Siegel’s lemma
  • Elliptic curve
  • j-invariant
  • Isogeny
  • Modular curve
  • Geodesics

Mathematics Subject Classification

  • Primary: 11H06
  • 52C07
  • 11G05