Quasi-random multilinear polynomials

  • Gil Kalai
  • Leonard J. SchulmanEmail author


We consider multilinear Littlewood polynomials, polynomials in n variables in which a specified set of monomials U have ±1 coefficients, and all other coefficients are 0. We provide upper and lower bounds (which are close for U of degree below log n) on the minimum, over polynomials h consistent with U, of the maximum of |h| over ±1 assignments to the variables. (This is a variant of a question posed by Erdős regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale–Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than 1; and on Bernstein’s approximation theory inequality.


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  1. [1]
    B. Berger, The fourth moment method, SIAM Journal on Computing 26 (1997), 1188–1207.MathSciNetzbMATHGoogle Scholar
  2. [2]
    A. Bonami, Étude des coefficients Fourier des fonctions de Lp(G), Université de Grenoble. Annales de l’Institut Fourier 20 (1970), 335–402.Google Scholar
  3. [3]
    P. Borwein, Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, Vol. 10, Springer-Verlag, New York, 2002.Google Scholar
  4. [4]
    T. A. Brown and J. H. Spencer, Minimization of ±1 matrices under line shifts, Colloquium Mathematicum 23 (1971), 165–171.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Carbery and J. Wright, Distributional and Lq norm inequalities for polynomials over convex bodies in Rn, Mathematical Research Letters 8 (2001), 233–248.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C. Carlet and S. Mesnager, Four decades of research on bent functions, Designs, Codes and Cryptography 73 (2016), 5–50.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. M. Childs, R. Kothari, M. Ozols and M. Roetteler, Easy and hard functions for the boolean hidden shift problem, in 8th Conference on the Theory of Quantum Computation, Communication and Cryptography, Leibniz International Proceedings in Informatics, Vol. 22, Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, Wadern, 2013, pp. 50–79.zbMATHGoogle Scholar
  8. [8]
    F. R.K. Chung, Quasi-random classes of hypergraphs, Random Structures & Algorithms 1 (1990), 363–382.MathSciNetCrossRefGoogle Scholar
  9. [9]
    F. R. K. Chung and R. L. Graham, Quasi-random hypergraphs, Random Structures & Algorithms 1 (1990), 105–124.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. R. K. Chung, R. L. Graham and R. M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. Conlon, H. Hàn, Y. Person and M. Schacht, Weak quasi-randomness for uniform hypergraphs, Random Structures & Algorithms 40 (2012), 1–38.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. F. Dillon, A survey of bent functions, NSA Technical Journal Special Issue, 1972, 191–215.Google Scholar
  13. [13]
    J. F. Dillon, Elementary Hadamard difference sets, PhD thesis, University of Maryland, College Park, 1974.zbMATHGoogle Scholar
  14. [14]
    I. Dinur, E. Friedgut, G. Kindler and R. O’Donnell, On the Fourier tails of bounded functions over the discrete cube, Israel Journal of Mathematics 160 (2007), 389–412.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    T. Erdélyi, Polynomials with Littlewood-type coefficient constraints, in Approximation Theory X, (St. Louis, MO, 2001), Innovations in Applied Mathematics, Vanderbilt University Press, Nashville, TN, 2002, pp. 153–196.zbMATHGoogle Scholar
  16. [16]
    P. Erdös, Some unsolved problems, Michigan Mathematical Journal 4 (1957), 291–300.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P. Erdös, An inequality for the maximum of trigonometric polynomials, Annales Polonici Mathematici 12 (1962), 151–154.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. C. Fishburn and N. J. A. Sloane, The solution to Berlekamp’s switching game, Discrete Mathematics 74 (1989), 263–290.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    P. Frankl, V. Rödl and R. M. Wilson, The number of submatrices of a given type in a Hadamard matrix and related results, Journal of Combinatorial Theory, Series B 44 (1988), 317–328.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Y. Gordon and H. S. Witsenhausen, On extensions of the Gale–Berlekamp switching problem and constants of p-spaces, Israel Journal of Mathematics 11 (1972), 216–229.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Haviland and A. G. Thomason, Pseudo-random hypergraphs, Discrete Mathematics 75 (1989), 255–278.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Jedwab, D. J. Katz and K.-U. Schmidt, Littlewood polynomials with small L4 norm, Advances in Mathematics 59 (2013), 3210–3214.Google Scholar
  23. [23]
    J.-P. Kahane, Sur les sommes vectorielles ±un, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 259 (1964), 2577–2580.zbMATHGoogle Scholar
  24. [24]
    J.-P. Kahane, Sur les polynômes á coefficients unimodulaires, Bulletin of the London Mathematical Society 12 (1980), 321–342.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Khintchine, Über dyadische Brüche, Mathematische Zeitschrift 18 (1923), 109–116.CrossRefGoogle Scholar
  26. [26]
    Y. Kohayakawa, V. Rödl, and J. Skokan, Hypergraphs, quasi-randomness, and conditions for regularity, Journal of Combinatorial Theory, Series A 97 (2002), 307–352.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Krawtchouk, Sur une généralisation des polynômes d’Hermite, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 189 (1929), 620–622.zbMATHGoogle Scholar
  28. [28]
    M. Krivelevich and B. Sudakov, Pseudo-random graphs, in More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies, Vol. 15, Springer, Berlin, 2006, pp. 199–262.Google Scholar
  29. [29]
    R. Latala and K. Oleszkiewicz, On the best constant in the Khintchine–Kahane inequality, Studia Mathematica 109 (1994), 101–104.MathSciNetzbMATHGoogle Scholar
  30. [30]
    V. I. Levenshtein, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Transactions on Information Theory 41 (1995), 1303–1321.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J. E. Littlewood, On polynomials±zm andeαmizm, z = eθi, Journal of the London Mathematical Society 41 (1966), 367–376, 1966. (Reprinted in The Collected Papers of J. E. Littlewood, Vol. 2, Oxford University Press, New York, 1982, pp. 1423-1433).Google Scholar
  32. [32]
    J. E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, D. C. Heath and Co. Raytheon Education, Lexington, MA, 1968.zbMATHGoogle Scholar
  33. [33]
    L. Lovász, Large Networks and Graph Limits, American Mathematical Society Colloquium Publications, Vol. 60, American Mathematical Society, Providence, RI, 2012.Google Scholar
  34. [34]
    R. Meka, O. Nguyen and V. Vu, Anti-concentration for polynomials of independent random variables, Theory of Computing 12 (2016), 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    P.-A. Meyer, Martingales and Stochastic Integrals. I, Lecture Notes in Mathematics, Vol. 284, Springer-Verlag, Berlin–New York, 1972.Google Scholar
  36. [36]
    I. P. Natanson, Constructive Theory of Functions. Vol. 1, Frederick Ungar Publishing, New York 1964.Google Scholar
  37. [37]
    R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.CrossRefzbMATHGoogle Scholar
  38. [38]
    R. E. A. C. Paley and A. Zygmund, A note on analytic functions in the unit circle, Proceedings of the Cambridge Philosophical Society 28 (1932), 266–272.CrossRefGoogle Scholar
  39. [39]
    R. K. Pathria and P. D. Beale, Statistical Mechanics, Academic Press, New York, 2011.zbMATHGoogle Scholar
  40. [40]
    H. Queffelec and B. Saffari, On Bernstein’s inequality and Kahane’s ultraflat polynomials, Journal of Fourier Analysis and Applications 2 (1996), 519–582.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    V. Rödl, On the universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986), 125–134.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    O. S. Rothaus, On “bent” functions, Journal of Combinatorial Theory. Series A 20 (1976), 300–305.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    W. Rudin, Some theorems on Fourier coefficients, Proceedings of the American Mathematical Society 10 (1959), 855–859.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    K.-U. Schmidt, An extremal problem for polynomials, Comptes Rendus Mathématique. Académie des Sciences 352 (2014), 95–97.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    H. S. Shapiro, Extremal problems for polynomials and power series, Master’s thesis, Massachusetts Institute of Technology, 1951.Google Scholar
  46. [46]
    M. Simonovits and V. T. Sós, Szemerédi’s partition and quasirandomness, Random Structures & Algorithms 2 (1991), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    A. Ta-Shma, Explicit, almost optimal, epsilon-balanced codes, in Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 2017, pp. 238–251.zbMATHGoogle Scholar
  48. [48]
    M. Talagrand, Majorizing measures: the generic chaining, Annals of Probability 24 (1996), 1049–1103.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    A. Thomason, Pseudorandom graphs, in Random Graphs’ 85 (Poznań, 1985), North-Holland Mathematics Studies, Vol. 144, North-Holland, Amsterdam, 1987, pp. 307–331.Google Scholar
  50. [50]
    A. Thomason, Random graphs, strongly regular graphs and pseudo-random graphs, in Surveys in Combinatorics 1987 (New Cross, 1987), London Mathematical Society Lecture Note Series, Vol. 123, Cambridge University Press, Cambridge, 1987, pp. 173–195.zbMATHGoogle Scholar
  51. [51]
    N. Tokareva, Bent Functions: Results and Applications to Cryptography, Academic Press, Amsterdam, 2015.CrossRefzbMATHGoogle Scholar
  52. [52]
    E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Annals of Mathematics, 62 (1955), 548–564.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael
  2. 2.Division of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadenaUSA

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