Abstract
Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products\(\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}\) integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.
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References
F. V. Atkinson, On a problem of Erdös and Szekeres, Canad. Math. Bull. 4 (1961), 7–12.
S. Chowla, The Riemann zeta and allied functions, Bull. Amer. Math. Soc. 58 (1952), 287–305.
S. Chowla, Some applications of a method of A. Selberg, J. Reine Angew. Math. 217 (1965), 128–132.
P. Erdös and G. Szekeres, On the product Πk =1 n(1 - z a k), Publ. de l’Institut mathematique, 1950.
Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms to the circle, Ergodic Theory Dynam. Systems 9 (1989), 643–680.
M. N. Kolountzakis, On nonnegative cosine polynomials with nonnegative integral coefficients, Proc. Amer. Math. Soc. 120 (1994), 157–163.
M. N. Kolountzakis, A construction related to the cosine problem, Proc. Amer. Math. Soc. 122 (1994), 1115–1119.
M. N. Kolountzakis, Some applications of probability to additive number theory and harmonic analysis, Number Theory (New York Seminar, 1991-1995), Springer, New York, 1996, pp. 229–251.
M. N. Kolountzakis, The density of Bh[g] sets and the minimum of dense cosine sums, J. Number Theory 56 (1996), 4–11.
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Theorie des Nombres de Bordeaux, 5 (1993), 411–484.
A. M. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc. (2), 26 (1982), 412–420.
G. Pisier, Arithmetic Characterization of Sidon Sets, Bull. Amer. Math. Soc. 8 (1983), 87–89.
W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227.
I. Z. Ruzsa, Negative values of cosine sums, Acta Arith. 111 (2004), 179–186.
A. Schinzel, On the number of irreducible factors of a polynomial, Topics in Number Theory, North Holland, Amsterdam, 1976, pp. 305–314.
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Partially supported by NSF Grant DMS 1301608
Partially supported by NSF Grant DMS 1764081
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Bourgain, J., Chang, MC. On a paper of Erdös and Szekeres. JAMA 136, 253–271 (2018). https://doi.org/10.1007/s11854-018-0060-9
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DOI: https://doi.org/10.1007/s11854-018-0060-9