Siegel’s Lemma Is Sharp

  • József Beck


Siegel’s Lemma is concerned with finding a “small” nontrivial integer solution of a large system of homogeneous linear equations with integer coefficients, where the number of variables substantially exceeds the number of equations (for example, n equations and N variables with N ≥ 2n), and “small” means small in the maximum norm. Siegel’s Lemma is a clever application of the Pigeonhole Principle, and it is a pure existence argument. The basically combinatorial Siegel’s Lemma is a key tool in transcendental number theory and diophantine approximation. David Masser (a leading expert in transcendental number theory) asked the question whether or not the Siegel’s Lemma is best possible. Here we prove that the so-called “Third Version of Siegel’s Lemma” is best possible apart from an absolute constant factor. In other words, we show that no other argument can beat the Pigeonhole Principle proof of Siegel’s Lemma (apart from an absolute constant factor). To prove this, we combine a concentration inequality (i.e., Fourier analysis) with combinatorics.



I am grateful to David Masser (Basel) who called my attention to the problem.


  1. 1.
    N. Alon, D.N. Kozlov, Coins with arbitrary weights. J. Algorithms 25, 162–176 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    N. Alon, V.H. Vu, Anti-Hadamard matrices, coin weighing, threshold gates and indecomposable hypergraphs. J. Comb. Theory Ser. A 79, 133–160 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. Baker, Transcendental Number Theory (Cambridge University Press, Cambridge, 1975)CrossRefMATHGoogle Scholar
  4. 4.
    E. Bombieri, W. Gubler, Heights in Diophantine Geometry. New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006)Google Scholar
  5. 5.
    E. Bombieri, J.D. Vaaler, On Siegel’s lemma. Invent. Math. 73(1), 11–32 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    P. Erdős, On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51, 898–902 (1945)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    G. Halász, Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hung. 8, 197–211 (1977)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A.M. Macbeath, On measure of sum-sets, II. The sum-theorem for the torus. Proc. Camb. Philos. Soc. 49, 40–43 (1953)CrossRefMATHGoogle Scholar
  9. 9.
    W.M. Schmidt, Diophantine Approximations and Diophantine Equations. Lecture Notes in Mathematics, vol. 1467 (Springer, Berlin, 1991)Google Scholar
  10. 10.
    J. Spencer, Six standard deviations suffice. Trans. Am. Math. Soc. 289, 679–706 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    J.D. Vaaler, The best constant in Siegel’s lemma. Monatshaft. Math. 140(1), 71–89 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    J.D. Vaaler, A.J. van der Poorten, Bounds for solutions of systems of linear equations. Bull. Aust. Math. Soc. 25, 125–132 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentRutgers UniversityNew BrunswickUSA

Personalised recommendations