On sparse geometry of numbers


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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  1. 1.

    Aliev, I., Averkov, G., De Loera, J.A., Oertel, T.: Optimizing sparsity over lattices and semigroups. In: Bienstock, D., Zambelli, G. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45771-6_4

    Google Scholar 

  2. 2.

    Aliev, I., De Loera, J., Oertel, T., O’Neill, C.: Sparse solutions of linear diophantine equations. SIAM J. Appl. Algebra. Geomet. 1(1), 239–253 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Baker, R., Masser, D.: Siegel’s lemma is sharp for almost all linear systems. Bull. Lond. Math. Soc. 51(5), 853–867 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bombieri, E., Vaaler, J.D.: On Siegel’s lemma. Invent. Math. 73(1), 11–32 (1983)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cohen, H., Strömberg, F.: Modular Forms. A Classical Approach. Graduate Studies in Mathematics, vol. 179. American Mathematical Society, Providence, RI (2017)

    Google Scholar 

  6. 6.

    Eldar, Y.C., Kutyniok, G.: Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  7. 7.

    Fukshansky, L., Needell, D., Sudakov, B.: An algebraic perspective on integer sparse recovery. Appl. Math. Comput. 340, 31–42 (2019)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Fukshansky, L., Guerzhoy, P., Luca, F.: On arithmetic lattices in the plane. Proc. Am. Math. Soc. 145(4), 1453–1465 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. North-Holland Publishing Co., New York (1987)

    Google Scholar 

  10. 10.

    Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen. Springer-Verlag, Berlin (1998)

    Google Scholar 

  11. 11.

    Kühnlein, S.: Well-rounded sublattices. Int. J. Numer. Theory. 8(5), 1133–1144 (2012)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Miyake, T.: Modular Forms Modular forms, Translated from the 1976 Japanese original by Yoshitaka Maeda. Springer Monographs in Mathematics. Springer, Berlin (2006)

    Google Scholar 

  13. 13.

    Schmidt, W.M.: Diophantine Approximations and Diophantine Equations. Lecture Notes in Mathematics, vol. 1467. Springer, Berlin (1991)

    Google Scholar 

  14. 14.

    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986)

    Google Scholar 

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

Mathematics Subject Classification

  • Primary: 11H06
  • 52C07
  • 11G05