×

Constants of the Kahane-Salem-Zygmund inequality asymptotically bounded by 1. (English) Zbl 07437673

Summary: The Kahane-Salem-Zygmund inequality for multilinear forms in \(\ell_\infty\) spaces claims that, for all positive integers \(m, n_1, \ldots, n_m\), there exists an \(m\)-linear form \(A : \ell_\infty^{n_1} \times \cdots \times \ell_\infty^{n_m} \longrightarrow \mathbb{K}\) (\(\mathbb{K} = \mathbb{R}\) or \(\mathbb{C}\)) of the type \[ A( z^{( 1 )}, \ldots, z^{( m )}) = \sum_{j_1 = 1}^{n_1} \cdots \sum_{j_m = 1}^{n_m} \pm z_{j_1}^{( 1 )} \cdots z_{j_m}^{( m )}, \] satisfying \[ \| A \| \leq C_m \max \{ n_1^{1 / 2} , \ldots , n_m^{1 / 2} \} \prod_{j = 1}^m n_j^{1 / 2}, \] for \[ C_m \leq \kappa \sqrt{ m \log m} \sqrt{ m !} \] and a certain \(\kappa > 0\). Our main result shows that given any \(\epsilon > 0\) and any positive integer \(m\), there exists a positive integer \(N\) such that \[ C_m < 1 + \epsilon, \] when we consider \(n_1, \ldots, n_m > N\). In addition, while the original proof of the Kahane-Salem-Zygmund relies on highly non-deterministic arguments, our approach is constructive. We also provide the same asymptotic bound (which is shown to be optimal in some cases) for the constant of a related non-deterministic inequality proved by G. Bennett in 1977. Applications to Berlekamp’s switching game are given.

MSC:

47A07 Forms (bilinear, sesquilinear, multilinear)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A51 Stochastic matrices (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J., Sharp generalizations of the multilinear Bohnenblust-Hille inequality, J. Funct. Anal., 266, 6, 3726-3740 (2014) · Zbl 1319.46035
[2] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J., Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators, Isr. J. Math., 211, 1, 197-220 (2016) · Zbl 1342.26040
[3] Albuquerque, N.; Rezende, L., Asymptotic estimates for unimodular multilinear forms with small norms on sequence spaces, Bull. Braz. Math. Soc., 52, 23-39 (2021) · Zbl 1470.46069
[4] Alon, N.; Spencer, J. H., The Probabilistic Method, Wiley Series in Discrete Mathematics and Optimization (2016), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. Hoboken, NJ, xiv+375 pp. · Zbl 1333.05001
[5] Araújo, G.; Pellegrino, D., A Gale-Berlekamp permutation-switching problem in higher dimensions, Eur. J. Comb., 77, 17-30 (2019) · Zbl 1411.91154
[6] Bayart, F., Maximum modulus of random polynomials, Q. J. Math., 63, 1, 21-39 (2012) · Zbl 1248.46033
[7] Beck, J.; Spencer, J., Balancing matrices with line shifts, Combinatorica, 3, 299-304 (1983) · Zbl 0542.05015
[8] Bennett, G.; Goodman, V.; Newman, C. M., Norms of random matrices, Pac. J. Math., 59, 2, 359-365 (1975) · Zbl 0325.47018
[9] Bennett, G., Schur multipliers, Duke Math. J., 44, 3, 603-639 (1977) · Zbl 0389.47015
[10] Boas, H. P., The football player and the infinite series, Not. Am. Math. Soc., 44, 11, 1430-1435 (1997) · Zbl 0909.30001
[11] Boas, H. P., Majorant series, Several Complex Variables. Several Complex Variables, Seoul, 1998. Several Complex Variables. Several Complex Variables, Seoul, 1998, J. Korean Math. Soc., 37, 2, 321-337 (2000) · Zbl 0965.32001
[12] Boas, H. P.; Khavinson, D., Bohr’s power series theorem in several variables, Proc. Am. Math. Soc., 125, 10, 2975-2979 (1997) · Zbl 0888.32001
[13] Bohnenblust, H. F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. Math., 32, 600-622 (1931) · JFM 57.0266.05
[14] Brown, T. A.; Spencer, J. H., Minimization of ±1 matrices under line shifts, Colloq. Math., 23, 165-171 (1971) · Zbl 0222.05016
[15] Brualdi, R. A.; Meyer, S. A., A Gale-Berlekamp permutation-switching problem, Eur. J. Comb., 44, 43-56 (2015) · Zbl 1321.91012
[16] Carl, B.; Maurey, B.; Puhl, J., Grenzordnungen von absolut-\((r, p)\)-summierenden Operatoren, Math. Nachr., 82, 205-218 (1978), (German) · Zbl 0407.47010
[17] Carlson, J.; Stolarski, D., The correct solution to Berlekamp’s switching game, Discrete Math., 287, 145-150 (2004) · Zbl 1054.94023
[18] Defant, A.; García, D.; Maestre, M., Maximum moduli of unimodular polynomials, Satellite Conference on Infinite Dimensional Function Theory. Satellite Conference on Infinite Dimensional Function Theory, J. Korean Math. Soc., 41, 1, 209-229 (2004) · Zbl 1050.32003
[19] Defant, A.; García, D.; Maestre, M.; Sevilla-Peris, P., Dirichlet Series and Holomorphic Functions in High-Dimensions, New Mathematical Monographs, vol. 37 (2019), Cambridge University Press · Zbl 1460.30004
[20] Defant, A.; Mastyło, M., Subgaussian Kahane-Salem-Zygmund inequalities in Banach spaces (2020), preprint, 47 pp.
[21] Fishburn, P. C.; Sloane, N. J.A., The solution to Berlekamp’s switching game, Discrete Math., 74, 263-290 (1989) · Zbl 0664.94024
[22] Horadam, K. J., Hadamard Matrices and Their Applications (2007), Princeton University Press · Zbl 1145.05014
[23] Kahane, J.-P., Some Random Series of Functions (1985), Cambridge University Press · Zbl 0571.60002
[24] Kalai, G.; Schulman, L. J., Quasi-random multilinear polynomials, Isr. J. Math., 230, 1, 195-211 (2019) · Zbl 1454.26018
[25] Mantero, A.; Tonge, A., The Schur multiplication in tensor algebras, Stud. Math., 68, 1, 1-24 (1980) · Zbl 0445.46051
[26] Mastyło, M.; Szwedek, R., Kahane-Salem-Zygmund polynomial inequalities via Rademacher processes, J. Funct. Anal., 272, 11, 4483-4512 (2017) · Zbl 1379.46015
[27] Pellegrino, D.; Serrano-Rodríguez, D.; Silva, J., On unimodular forms with small norms on sequence spaces, Linear Algebra Appl., 595, 24-32 (2020) · Zbl 1440.15016
[28] Pellegrino, D.; Silva, J.; Teixeira, E., On a continuous Gale-Berlekamp switching game, An. Acad. Bras. Ciênc. (2021), in press
[29] Santos, J.; Velanga, T., On the Bohnenblust-Hille inequality for multilinear forms, Results Math., 72, 1-2, 239-244 (2017) · Zbl 06785696
[30] Spencer, J., Discrete Ham Sandwich theorems, Eur. J. Comb., 2, 291-298 (1981) · Zbl 0476.05025
[31] Spencer, J., Ten Lectures on the Probabilistic Method, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 64 (1994), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0822.05060
[32] Varopoulos, N. Th., On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Funct. Anal., 16, 83-100 (1974) · Zbl 0288.47006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.