×

Average growth of \(L_p\) norms of Erdős-Szekeres polynomials. (English) Zbl 1513.42105

Let \( A_p(M,n) \) be the average of \( \vert P_n(\{s_j\},\cdot)\Vert_p^p \) over all \( n- \)tuples of positive integers \( s_j\leq M, \) where \[ P_n(\{s_j\},z)=\prod_{j=1}^n(1-z^{s_j}), \] and \( \Vert\cdot\Vert_p \) is the normalized \( p- \)norm on the unit circle \( \vert z\vert =1. \) The corresponding variance is denoted by \( V_p(M,n). \) The authors study the asymptotic behaviour of those quantities in different regimes. For example, Theorems 1.2 and 1.4 give the value of the limit \( \lim_{k\to\infty}A_p(M_k,n_k)^{1/n_k}\), \(p\geq 1 \) and \(\lim_{k\to\infty}V_p(M_k,n_k)^{1/n_k}\), \(p\geq 2\), \(\rho=1, \) where \( \{M_k,\},\{n_k\} \) are sequences of positive integers with limit \( \infty \) such that for some \( \rho\in[1,\infty]\quad \lim_{k\to\infty}M_k^{1/n_k}=\rho. \) The expressions for the limits are essentially simpler for the case \( p=2 \) (Theorem 1.2). For the fixed \( n \) the limits \( \lim_{M\to\infty}A_p(M,n) \) and \( \lim_{M\to\infty}V_p(M,n)=0 \) are found for \( p>0 \) (Theorem 1.3). Also the averages over subsequences of the integers that generate uniformly distributed subsequences, rather than requiring all \( 1\leq s_j\leq M, \) and varying bounds instead of the uniform bound \( M \) are considered.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
11C08 Polynomials in number theory
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Aistleitner, R. Hofer, and G. Larcher, On parametric thue-morse sequences and lacunary trigonometric products, Ann. Inst. Fourier, 67 (2017), 637-687. · Zbl 1437.11042
[2] C. Aistleitner, G. Larcher, F. Pillichshammer, S. Saad Eddin, and R. F. Tichy, On Weyl products and uniform distribution modulo one, Monatsh. Math., 185 (2018), 363-395. · Zbl 1410.11106
[3] C. Aistleitner, N. Technau, and A. Zafeiropoulos, On the order of magnitude of Sudler products, arXiv: 2002.06602v1. · Zbl 1451.11082
[4] A. Avila, S. Jitomirskaya, and C. A. Marx, Spectral theory of extended Harper’s model and a question by Erdős and Szekeres, Invent. Math., 210 (2017), 2697-2718. · Zbl 1380.37019
[5] J. P. Bell, P. B. Borwein and L. B. Richmond, Growth of the product \(\prod_{j=1}^n(1-x^{a_j})\), Acta Arith., 86 (1998), 155-170. · Zbl 0918.11054
[6] A. S. Belov and S. V. Konyagin, An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients, Izv. Math., 60 (1996), 1123-1182. · Zbl 0885.42001
[7] C. Billsborough, M. J. Freedman, S. Hart, G. Kowalsky, D. S. Lubinsky, A. Pomeranz, and A. Sammel, On lower bounds for Erdös-Szekeres polynomials, Proc. Amer. Math. Soc., 149 (2021), 4233-4246. · Zbl 1471.42056
[8] P. Borwein, Some restricted partition functions, J. Number Theory, 45 (1993), 228- 240. · Zbl 0788.11043
[9] P. Borwein and C. Ingalls, The Prouhet-Tarry-Escott problem revisited, Enseign. Math., 49 (1994), 3-27. · Zbl 0810.11016
[10] J. Bourgain and Mei-Chu Chang, On a paper of Erdős and Szekeres, J. Anal. Math., 136 (2018), 253-271. · Zbl 1453.11040
[11] P. Erdős, Problems and results on Diophantine approximation, Compos. Math., 16 (1964), 52-66. · Zbl 0131.04803
[12] P. Erdős and G. Szekeres, On the product \(\prod_{k=1}^n( 1-z^{a_k}) \), Acad. Serbe Sci. Publ. Inst. Math., 13 (1959), 29-34. · Zbl 0097.03302
[13] S. Grepstad, L. Kaltenböck, and M. Neumüller, A positive lower bound for \(\liminf_{N\rightarrow \infty} \prod_{r=1}^N\vert 2\sin \pi r\phi \vert \), Proc. Amer. Math. Soc., 147 (2019), 4863-4876. · Zbl 1448.11138
[14] S. Greptsad and M. Neumüller, Asymptotic behavior of the Sudler product of sines for quadratic irrationals, J. Math. Anal. Appl., 465 (2018), 928-960. · Zbl 1415.40004
[15] S. Grepstad, L. Kaltenböck, and M. Neumüller, On the asymptotic behavior of the sine product \(\prod_{r=1}^N\left\vert 2\sin \pi r\alpha \right\vert \), arXiv:1909.00980, to appear in Discrepancy Theory (eds. D. Bilyk, J. Dick, F. Pillichshammer). · Zbl 1459.11017
[16] H. Niederreiter and L. Kuipers, Uniform Distribution of Sequences, Wiley (New York, 1974). · Zbl 0281.10001
[17] D. S. Lubinsky, The size of \(( q;q)_n\) for q on the unit circle, J. Number Theory, 76 (1999), 217-247. · Zbl 0940.11007
[18] R. Maltby, Root systems and the Erdős-Szekeres problem, Acta Arith., 81 (1997), 229-245. · Zbl 0881.11030
[19] R. Maltby, Pure product polynomials and the Prouhet-Tarry-Escott problem, Math. Comp., 219 (1997), 1323-1340. · Zbl 1036.11539
[20] C. Sudler, An estimate for a restricted partition function, Quart. J. Math. Oxford, 15 (1964), 1-10. · Zbl 0151.01402
[21] P. Verschueren and B. Mestel, Growth of the Sudler product of sines at the golden rotation number, J. Math. Anal. Appl., 433 (2016), 200-226. · Zbl 1335.11018
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.