A Class of Littlewood Polynomials that are Not L^α-Flat

[1] EL ABDALAOUI, E. H.: Ergodic Banach problem, flat polynomials and Mahler’s measures with combinatorics, preprint 2016, arXiv:1508.06439v5 [math.DS].Search in Google Scholar

[2] EL ABDALAOUI, E. H.—NADKARNI, M.: Some notes on flat polynomials, preprint 2014, http://arxiv.org/abs/1402.5457Search in Google Scholar

[3] EL ABDALAOUI, E. H.—NADKARNI, M.: On flat polynomials with non-negative coefficients, preprint 2015, https://arxiv.org/abs/1508.00417Search in Google Scholar

[4] BALISTER, P.—BOLLOBÁS, B.—MORRIS, R.—SAHASRABUDHE, J.—TIBA, M.: Flat Littlewood polynomials exist, arXiv:1907.09464v1 [math.CA].Search in Google Scholar

[5] BECK, J.: Flat polynomials on the unit circle - Note on a Problem of Littlewood, Bull. London Math. Soc., 23 (1991), 269–277.10.1112/blms/23.3.269Search in Google Scholar

[6] BOMBIERI, E.—BOURGAIN, J.: On Kahane’s ultraflat polynomials, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 627–703.10.4171/JEMS/163Search in Google Scholar

[7] BORWEIN, P.—FERGUSON, R.—KNAUER, J.: The merit factor problem, In: The Merit Factor Problem. Number Theory and Polynomials, London Math. Soc. Lecture Note Ser., Vol. 352, Cambridge Univ. Press, Cambridge, 2008, 52–70,10.1017/CBO9780511721274.006Search in Google Scholar

[8] CHUNG, K. L.: A Course in Probability Theory, 3rd. ed. Academic Press, Inc., San Diego, CA, 2001.Search in Google Scholar

[9] DOWNAROWICZ, T.—LACROIX, Y.: Merit factors and Morse sequences, Theoretical Computer Science, (209) (1998), 377–387.10.1016/S0304-3975(98)00121-2Search in Google Scholar

[10] ERDŐS, P.: An inequality for the maximum of trigonometric polynomials, Ann. Polon. Math., (12) (1962), 151–154.10.4064/ap-12-2-151-154Search in Google Scholar

[11] ERDŐS, P.: Some unsolved problems, Michigan Math. J. 4 (1957), 291–300.10.1307/mmj/1028997963Search in Google Scholar

[12] ERDŐS, P.: Problems and results on polynomials and interpolation.In: (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), Aspects of Contemporary Complex analysis Academic Press, London-New York, 1980, pp. 383–391,Search in Google Scholar

[13] FUKUYAMA, K.: A central limit theorem for trigonometric series with bounded gaps, Probab. Theory Relat. Fields, 149 (2011), 139–148.10.1007/s00440-009-0245-3Search in Google Scholar

[14] HØHOLDT, T.: The merit factor problem for binary sequences, In: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Lecture Notes in Comput. Sci. Vol. 3857, Springer-Verlag, Berlin, 2006, pp. 51–59.10.1007/11617983_4Search in Google Scholar

[15] JENSEN, H. E.—HØHOLDT, T.: Determination of the merit factor of Legendre sequences, IEEE Trans. Inform. Theory, 34 (1988), no. 1, 161–164.10.1109/18.2620Search in Google Scholar

[16] JENSEN, J.—JENSEN, H. E.—HØHOLDT, T.: The merit factor of binary sequences related to difference sets,IEEE Trans. Inform.Theory, 37 1991, no. 3, 617–626.10.1109/18.79917Search in Google Scholar

[17] GOLAY, M. J. E.: Sieves for low autocorrelation binary sequences, IEEE Trans. Inform. Theory, 23 (1977), no. 1, pp. 43–51.10.1109/TIT.1977.1055653Search in Google Scholar

[18] JEDWAB, J.: A survey of the merit factor problem for binary sequences. In: Third international conference, Seoul, Korea, October 24–28, 2004. (T. Helleseth, ed. et al.), Sequences and Their Applications SETA–2004, Revised selected papers. Lecture Notes in Computer Science Vol. 3486, Springer-Verlag, Berlin 2005, pp. 30–55.Search in Google Scholar

[19] KAHANE, J-P.: Sur les polynômesà coefficients unimodulaires, Bull. London. Math. Soc. 12 (1980), 321–342.10.1112/blms/12.5.321Search in Google Scholar

[20] LITTLEWOOD, J. E.: On polynomials∑n±zm,∑neαmizm,z=eθi\sum\limits_{}^n { \pm {z^m}} ,\sum\limits_{}^n {{e^{{\alpha _m}i}}{z^m}} ,z = {e^{\theta i}} J. London Math. Soc. 41 (1966), 367–376.10.1112/jlms/s1-41.1.367Search in Google Scholar

[21] LITTLEWOOD, J. E.: Some Problems in Real and Complex Analysis. D.C.Heath and Co. Raytheon Education Co., Lexington, Mass. 1968.Search in Google Scholar

[22] LITTLEWOOD, J. E.: On the mean values of certain trigonometric polynomials,J. London Math. Soc. 36, 1961, 307–334.10.1112/jlms/s1-36.1.307Search in Google Scholar

[23] LITTLEWOOD, J. E.: On the mean values of certain trigonometric polynomials. II, Illinois J. Math. 6, 1962, 1–39.10.1215/ijm/1255631803Search in Google Scholar

[24] NEWMAN, D. J.: Norms of polynomials, Amer. Math. Monthly, 67 (1960), no. 8, pp. 778–779.10.2307/2308661Search in Google Scholar

[25] NEWMAN, D. J.—BYRNES, J. S.: The L⁴ norm of a polynomial with coefficients ±1, Amer. Math. Monthly 97 (1990), no 1,42–45.10.1080/00029890.1990.11995544Search in Google Scholar

[26] ODLYZKO, A. M.: Search for ultraflat polynomials with plus and minus one coefficients, In: (S. Butler ed. et al.) Connections in Discrete Mathematics. (A celebration of the work of Ron Graham), Cambridge University Press, Cambridge, 2018, pp. 39–5510.1017/9781316650295.004Search in Google Scholar

[27] QUEFFELEC, H.—SAFFARI, B.: On Bernstein’s inequality and Kahane’s ultraflat polynomials, J. Fourier Anal. Appl. 2 (1996), no. 6, 519–582.10.1007/s00041-001-4043-2Search in Google Scholar

[28] SAFFARI, B.—SMITH, B.: Inexistence de polynômes ultra-plats de Kahane à coefficients ±1. Preuve de la conjecture d’Erdős. (French) [Nonexistence of ultra-flat Kahane polynomials with coefficients ±1. Proof of the Erdős conjecture], C. R. Acad. Sci. Paris Sér I Math. 306 (1988), no. 16, 695–698.Search in Google Scholar

[29] FREDMAN, M.—SAFFARI, B.—SMITH, B.: Polynômes réciproques: conjecture d’Erdős en norme L⁴, taille des autocorrélations et inexistence des codes de Barker. (French) [Self-inversive polynomials: L⁴-norm Erd˝os conjecture, size of autocorrelations and nonexistence of Barker codes], C.R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 15, 461–464.Search in Google Scholar

[30] SCHILLING, L. R.: Measures, Integrals and Martingales. Second edition. Cambridge University Press, Cambridge, 2017.Search in Google Scholar

[31] ZYGMUND, A.: Trigonometric Series Vol. I & II. Second edition. Cambridge Univ. Press, Cambridge, 1959.Search in Google Scholar