Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 333–360 | Cite as

Threshold functions and Poisson convergence for systems of equations in random sets

  • Juanjo RuéEmail author
  • Christoph Spiegel
  • Ana Zumalacárregui


We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets, \(B_{h}[g]\)-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property “\(\mathcal {A}\) contains a non-trivial solution of \(M\cdot \mathbf{x }=\mathbf 0 \)” where \(\mathcal {A}\) is a random set and each of its elements is chosen independently with the same probability from the interval of integers \(\{1,\dots ,n\}\). Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.



The authors thank Christian Elsholtz for providing many references concerning linear systems of equations. We also thank Katy Beeler, Arnau Padrol and Lluis Vena for valuable input and fruitful discussions as well as Javier Cilleruelo for a detailed reading of the manuscript and support. Furthermore, we are thankful to the anonymous referees for detailed feedback and constructive suggestions. We would like to thank the Combinatorics and Graph Theory group at the Freie Universität Berlin where part of this research took place for working conditions and hospitality.


  1. 1.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd edn. Wiley, New York (2008)Google Scholar
  2. 2.
    Baltz, A., Hegarty, P., Knape, J., Larsson, U., Schoen, T.: The Structure of Maximum Subsets of \(\{1,\dots ,n\}\) with No Solutions to \(a+b=kc\). Electron. J. Combin. 12:Research Paper 19, 16 (2005)Google Scholar
  3. 3.
    Beck, M., Robins, S.: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Integer-Point Enumeration in Polyhedra. Springer, New York (2007)Google Scholar
  4. 4.
    Behrend, F.A.: On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. USA 32, 331–332 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bloom, T.F.: A quantitative improvement for Roth’s theorem on arithmetic progressions. J. London Math. Soc. 93(3), 643–663 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bollobás, B., Thomason, A.G.: Threshold functions. Combinatorica 7(1), 35–38 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J.: Roth’s theorem on progressions revisited. J. Anal. Math. 104, 155–192 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Büeler, B., Enge, A., Fukuda, K.: Exact volume computation for polytopes: a practical study. In: Polytopes—Combinatorics and Computation (Oberwolfach, 1997), vol 29 of DMV Sem, pp. 131–154. Birkhäuser, Basel (2000)Google Scholar
  9. 9.
    Chung, F.R.K., Goldwasser, J. L.: Integer sets containing no solutions to \(x+y=3k\). In: The Mathematics of Paul Erdős, pp. 267–277. Springer, Heidelberg (1996)Google Scholar
  10. 10.
    Chung, F.R.K., Goldwasser, J.L.: Maximum subsets of \((0,1]\) with no solutions to \(x+y=kz\). Electron. J. Comb. 3(1):Research Paper 1, approx. 23 pp (1996)Google Scholar
  11. 11.
    Cilleruelo, J.: Sidon sets in \(\mathbb{N}^d\). J. Comb. Theory Ser. A 117(7), 857–871 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cilleruelo, J., Ruzsa, I., Vinuesa, C.: Generalized Sidon sets. Adv. Math. 225(5), 2786–2807 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cilleruelo, J., Ruzsa, I.Z., Trujillo, C.: Upper and lower bounds for finite \(B_h[g]\) sequences. J. Number Theory 97(1), 26–34 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cilleruelo, J., Tesoro, R.: On sets free of sumsets with summands of prescribed size. Combinatorica (2017). doi: 10.1007/s00493-016-3444-4
  15. 15.
    De Loera, J.A.: The many aspects of counting lattice points in polytopes. Math. Semesterber. 52(2), 175–195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    De Loera, J. A., Rambau, J., Santos, F.: Triangulations, Volume 25 of Algorithms and Computation in Mathematics. Springer, Berlin (2010)Google Scholar
  17. 17.
    Ehrhart, E.: Sur les polyèdres homothétiques bordés à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 988–990 (1962)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Elkin, M.: An improved construction of progression-free sets. Israel J. Math. 184, 93–128 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Erdős, P., Rényi, A.: On the evolution of random graphs. In: Publication of the Mathematical Institute of the Hungarian Academy of Sciences, pp. 17–61 (1960)Google Scholar
  20. 20.
    Erdös, P., Turán, P.: On a problem of sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16, 212–215 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 31, 204–256 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Godbole, A.P., Janson, S., Locantore Jr., N.W., Rapoport, R.: Random Sidon sequences. J. Number Theory 75(1), 7–22 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gowers, W.T.: A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11(3), 465–588 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Green, B., Wolf, J.: A note on Elkin’s improvement of Behrend’s construction. In: Additive Number Theory, pp. 141–144. Springer, New York (2010)Google Scholar
  25. 25.
    Gunderson, D.S., Rödl, V.: Extremal problems for affine cubes of integers. Comb. Probab. Comput. 7(1), 65–79 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Heath-Brown, D.R.: Integer sets containing no arithmetic progressions. J. Lond. Math. Soc. (2) 35(3), 385–394 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Janson, S., Ruciński, A.: Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs. Arkiv für Matematik 49(1), 79–96 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kim, J.H., Vu, V.H.: Concentration of multivariate polynomials and its applications. Combinatorica 20(3), 417–434 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kohayakawa, Y., Lee, S.J., Rödl, V., Samotij, W.: The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers. Random Struct. Algorithms 46(1), 1–25 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lindström, B.: An inequality for \(B_{2}\)-sequences. J. Comb. Theory 6, 211–212 (1969)CrossRefzbMATHGoogle Scholar
  31. 31.
    Lyall, N.: A new proof of Sárközy’s theorem. Proc. Am. Math. Soc. 141, 2253–2264 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Macdonald, I.G.: The volume of a lattice polyhedron. Proc. Camb. Philos. Soc. 59, 719–726 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    O’Bryant, K.: Sets of integers that do not contain long arithmetic progressions. Electron. J. Comb. 18(1):Paper 59, 15 (2011)Google Scholar
  34. 34.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)zbMATHGoogle Scholar
  35. 35.
    Rödl, V., Ruciński, A.: Rado partition theorem for random subsets of integers. Proc. Lond. Math. Soc. (3) 74(3), 481–502 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Roth, K.F.: On certain sets of integers. J. Lond. Math. Soc. 1(1), 104–109 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ruciński, A.: Small subgraphs of random graphs—A survey. In: Random Graphs ’87 (Poznań, 1987), pp. 283–303. Wiley, Chichester (1990)Google Scholar
  38. 38.
    Ruzsa, I.Z.: Difference sets without squares. Period. Math. Hung. 15(3), 205–209 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ruzsa, I.Z.: Solving a linear equation in a set of integers I. Acta Arith. 65(3), 259–282 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ruzsa, I.Z.: Solving a linear equation in a set of integers II. Acta Arith. 72(4), 385–397 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Sándor, C.: Non-degenerate hilbert cubes in random sets. J. Théorie Nombres Bordeaux 19(1), 249–261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Sárközy, A.: On difference sets of sequences of integers. III. Acta Math. Acad. Sci. Hung. 31(3–4), 355–386 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  44. 44.
    Shapira, A.: Behrend-type constructions for sets of linear equations. Acta Arith. 122(1), 17–33 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Szemerédi, E.: On sets of integers containing no \(k\) elements in arithmetic progression. Acta Arith. 27:199–245 (1975) (Collection of articles in memory of Juriĭ Vladimirovič Linnik)Google Scholar
  46. 46.
    Warnke, L.: Upper tails for arithmetic progressions in random subsets. Israel J. Math. (To appear). arXiv:1612.08559
  47. 47.
    Ziegler, G.M.: Lectures on Polytopes Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Juanjo Rué
    • 1
    Email author
  • Christoph Spiegel
    • 1
  • Ana Zumalacárregui
    • 2
  1. 1.Department of MathematicsUniversitat Politècnica de Catalunya and Barcelona Graduate School of MathematicsBarcelonaSpain
  2. 2.Department of Pure MathematicsUniversity of New South WalesSydneyAustralia

Personalised recommendations