## ORTHOGONALITY AND THE MAXIMUM OF LITTLEWOOD COSINE POLYNOMIALS

Citations: | 1 - 0 self |

### BibTeX

@MISC{Erdélyi_orthogonalityand,

author = {Tamás Erdélyi},

title = {ORTHOGONALITY AND THE MAXIMUM OF LITTLEWOOD COSINE POLYNOMIALS},

year = {}

}

### OpenURL

### Abstract

Abstract. We prove that if p = 2q +1 is a prime, then the maximum of a Littlewood cosine polynomial qX Tq(t) = aj cos(jt), aj ∈ {−1, 1}, j=0 on the real line is at least c1 exp(c2(log q) 1/2), with an absolute constant c1 and c2 = p (log 2)/8. In the last section we observe that the maximum modulus of a Barker polynomial p of degree n on the unit circle of the complex plane is always at least √ n + p 1/3.

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Computational Excursions in Analysis and Number Theory
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(Show Context)
Citation Context ...ooks rather difficult. The result below stated in [2] is straightforward from [8, pages 285-288] which offers an elegant book proof of the Littlewood Conjecture first shown in [11] and [16]. The book =-=[1]-=- deals with a number of related topics. Littlewood [12, 13, 14, 15,] was interested in many closely related problems. Theorem 1.1. Let λ0 < λ1 < · · · < λm be nonnegative integers and let m∑ Sm(t) = A... |

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(Show Context)
Citation Context ...egree n with coefficients in {−1, 1}. Observe that if P ∈ Ln then the Parseval formula gives Hence ∫ π −π |P(e it )| 2 dt = 2π(n + 1) . max t∈[−π,π] |P(eit)| ≥ √ n + 1 for every P ∈ Ln. In 1957 Erdős =-=[9]-=- made the following conjecture. 2Conjecture 1.3. There is an absolute constant c > 0 such that for every P ∈ Ln. max t∈[−π,π] |P(eit)| ≥ (1 + c) √ n + 1 This is still a quite open problem today. Even... |

28 |
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(Show Context)
Citation Context ...Turyn and Storer showed that no even degree Barker polynomials exist for n > 12 (and indeed, as Schmidt [21] shows, none exist for any degree between 12 and 10 20 . It can also be shown (see Turyn 14=-=[25]-=-) that any odd degree Barker polynomial of degree greater than 12 must have degree of the form 4s 2 − 1, where s is an odd composite number.” In [3] the authors amend an argument of Saffari showing th... |

24 |
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(Show Context)
Citation Context ...“It is conjectured that no Barker polynomials exist for n > 12. See Saffari [19] and [20] for more about Barker polynomials and the proof of the nonexistence of self-reciprocal Barker polynomials. In =-=[23]-=- and [24] Turyn and Storer showed that no even degree Barker polynomials exist for n > 12 (and indeed, as Schmidt [21] shows, none exist for any degree between 12 and 10 20 . It can also be shown (see... |

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(Show Context)
Citation Context ...:= pk+1 − pk ≤ K(log pk) 2 with K := 2 exp(−γ) = 1.1229 . . . , where γ is the Euler constant. The probabilistic model behind the corrected form of Cramér’s conjecture is explained by A. Granville in =-=[10]-=-. See also Soundararajan’s survey [22]. Modulo the truth of Cramér’s conjecture, and even modulo the much weaker conjecture g(pk) = o(exp(c(log pk) 1/2 ) with c = √ (log 2)/8 Theorem 2.1 remains obvio... |

14 | On the mean values of certain trigonometric polynomials - Littlewood - 1961 |

14 | Cyclotomic integers and finite geometry
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(Show Context)
Citation Context ...mials and the proof of the nonexistence of self-reciprocal Barker polynomials. In [23] and [25] Turyn and Storer showed that no even degree Barker polynomials exist for n > 12 (and indeed, as Schmidt =-=[21]-=- shows, none exist for any degree between 12 and 10 20 . It can also be shown (see Turyn [24]) that any odd degree Barker polynomial of degree greater than 12 must have degree of the form 4s 2 − 1, wh... |

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Citation Context ...> 0 such that for every P ∈ Ln. max t∈[−π,π] |P(eit)| ≥ √ n + 1 + c However, as a consequence of Theorem 1.2 we can observe at least the following result, the derivation of which may also be found in =-=[6]-=-. Theorem 1.5. There is an absolute constant c > 0 such that for every P ∈ Ln. max t∈[−π,π] |P(eit)| ≥ √ n + c log n Proof of Theorem 1.5. Let P ∈ Ln. Observe that with Q(e it ) := |P(e it )| 2 = P(e ... |

10 |
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Citation Context ...und for (1.1) looks rather difficult. The result below stated in [2] is straightforward from [8, pages 285-288] which offers an elegant book proof of the Littlewood Conjecture first shown in [11] and =-=[16]-=-. The book [1] deals with a number of related topics. Littlewood [12, 13, 14, 15,] was interested in many closely related problems. Theorem 1.1. Let λ0 < λ1 < · · · < λm be nonnegative integers and le... |

10 |
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(Show Context)
Citation Context ...s for n > 12. Also ‖p‖L4(∂D) > √ n + 1 + 1 implies ‖p‖ L4(∂D) > ((n + 1) 2 + n + 1) 1/4 . In [1] P. Borwein writes “It is conjectured that no Barker polynomials exist for n > 12. See Saffari [19] and =-=[20]-=- for more about Barker polynomials and the proof of the nonexistence of self-reciprocal Barker polynomials. In [23] and [24] Turyn and Storer showed that no even degree Barker polynomials exist for n ... |

9 |
Sur le minimum d’une somme de cosinus
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(Show Context)
Citation Context .... It is difficult to estimate this minimum uniformly for every set of size n. Bourgain [4] proved minf(x) < −c1 exp(c2(log n) c3 ) with unspecified absolute constants c1, c2, and c3. In another paper =-=[5]-=- he showed that one can take c3 = 1/2 under the assumption that A ⊂ [1, n2 √ n ]. Our aim is to prove this without restriction. Theorem A. With the notations we have minf(x) < −c4 exp(c5(log n) 1/2 ) ... |

9 |
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(Show Context)
Citation Context ...njectured that no Barker polynomials exist for n > 12. See Saffari [19] and [20] for more about Barker polynomials and the proof of the nonexistence of self-reciprocal Barker polynomials. In [23] and =-=[24]-=- Turyn and Storer showed that no even degree Barker polynomials exist for n > 12 (and indeed, as Schmidt [21] shows, none exist for any degree between 12 and 10 20 . It can also be shown (see Turyn 14... |

7 | Lower bounds for the number of zeros of cosine polynomials: a problem of
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(Show Context)
Citation Context ...11P99; Secondary: 05D99. 1 Typeset by AMS-TEXobviously holds. But how large can (1.1) max t∈[−π,π] Sm(t) be? To give a decent lower bound for (1.1) looks rather difficult. The result below stated in =-=[2]-=- is straightforward from [8, pages 285-288] which offers an elegant book proof of the Littlewood Conjecture first shown in [11] and [16]. The book [1] deals with a number of related topics. Littlewood... |

7 |
Barker sequences and flat polynomials, Number Theory and Polynomials
- Borwein, Mossinghoff
(Show Context)
Citation Context ... 12 and 10 20 . It can also be shown (see Turyn 14[25]) that any odd degree Barker polynomial of degree greater than 12 must have degree of the form 4s 2 − 1, where s is an odd composite number.” In =-=[3]-=- the authors amend an argument of Saffari showing that Barker polynomials are flat. More precisely, if pn is a Barker polynomial of degree n, then α1 + O(1/n) ≤ |pn(z)| √ n ≤ α2 + O(1/n) for each z ∈ ... |

6 | Some applications of a method of A - Chowla - 1965 |

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4 |
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(Show Context)
Citation Context ...olynomials for n > 12. Also ‖p‖L4(∂D) > √ n + 1 + 1 implies ‖p‖ L4(∂D) > ((n + 1) 2 + n + 1) 1/4 . In [1] P. Borwein writes “It is conjectured that no Barker polynomials exist for n > 12. See Saffari =-=[19]-=- and [20] for more about Barker polynomials and the proof of the nonexistence of self-reciprocal Barker polynomials. In [23] and [24] Turyn and Storer showed that no even degree Barker polynomials exi... |

4 |
The distribution of prime numbers, Equidistribution in number theory, an introduction
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- 2007
(Show Context)
Citation Context ...e Problem in [6] below. 3swhere γ is the Euler constant. The probabilistic model behind the corrected form of Cramér’s conjecture is explained by A. Granville in [10]. See also Soundararajan’s survey =-=[22]-=-. Modulo the truth of Cramér’s conjecture, and even modulo the much weaker conjecture g(pk) = o(exp(c(log pk) 1/2 ) with c = √ (log 2)/8 a version of Theorem 2.1 remains obviously valid for all positi... |

4 |
Wieferich pairs and Barker sequences, Des
- Mossinghoff
(Show Context)
Citation Context ...n(z)| √ n ≤ α2 + O(1/n) for each z ∈ ∂D, where α1 = √ 1 − θ = 0.52477485 . . . and α2 = √ 1 + θ = 1.31324459 . . . , and sin θ := sup t>0 2 t t = 0.7246113537 . . . . In a recent work, M. Mossinghoff =-=[19]-=- showed that if a Barker sequence of length n > 13 exists, then either n = 189260468001034441522766781604 or n > 2 · 10 30 . In this section we record the following observation about the maximum modul... |

3 |
On a problem of Littlewood
- Konyagin
- 1981
(Show Context)
Citation Context ... lower bound for (1.1) looks rather difficult. The result below stated in [2] is straightforward from [8, pages 285-288] which offers an elegant book proof of the Littlewood Conjecture first shown in =-=[11]-=- and [16]. The book [1] deals with a number of related topics. Littlewood [12, 13, 14, 15,] was interested in many closely related problems. Theorem 1.1. Let λ0 < λ1 < · · · < λm be nonnegative intege... |

3 |
Negative values of cosine sums
- Ruzsa
(Show Context)
Citation Context ...mials Tq(t) = q∑ aj cos(jt), aj ∈ {−1, 1} , j=0 at least in the case when p = 2q + 1 is an odd prime. This is the content of our main result, Theorem 2.1. To this end we rely heavily on Ruzsa’s paper =-=[17]-=-, who claims the best result today to solve Chowla’s Cosine Problem in [6] below. 3Problem 1.6. Let A ⊂ N be a finite set of distinct integers and set ∑ m(A) := − min cos(at). t∈[−π,π] a∈A What is m(... |

2 |
On a problem of Littlewood, Mathematics of the USSR, Izvestia 18
- Konyagin
- 1981
(Show Context)
Citation Context ... lower bound for (1.1) looks rather difficult. The result below stated in [2] is straightforward from [8, pages 285-288] which offers an elegant book proof of the Littlewood Conjecture first shown in =-=[12]-=- and [17]. The book [1] deals with a number of related topics. Littlewood [13, 14, 15, 16,] was interested in many closely related problems. Theorem 1.1. Let λ0 < λ1 < · · · < λm be nonnegative intege... |

1 |
le minimum de certaines sommes de cosinus, in Harmonic analysis: study group on translation-invariant Banach spaces, Exp
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- 1984
(Show Context)
Citation Context ... a finite set of positive integers, |A| = n, and write f(x) = ∑ cos(ax). a∈A Since f(0) > 0, we have min f(x) < 0. It is difficult to estimate this minimum uniformly for every set of size n. Bourgain =-=[4]-=- proved minf(x) < −c1 exp(c2(log n) c3 ) with unspecified absolute constants c1, c2, and c3. In another paper [5] he showed that one can take c3 = 1/2 under the assumption that A ⊂ [1, n2 √ n ]. Our a... |

1 |
Wieferich pairs and Barker sequences (2009
- Mossinghoff
(Show Context)
Citation Context ...n(z)| √ n ≤ α2 + O(1/n) for each z ∈ ∂D, where α1 − √ 1 − θ = 0.52477485 . . . and α2 = √ 1 + θ = 1.31324459 . . . , and sin θ := sup t>0 2 t t = 0.7246113537 . . . . In a recent work, M. Mossinghoff =-=[18]-=- showed that if a Barker sequence of length n > 13 exists, then either n = 189260468001034441522766781604 or n > 2 · 10 30 . In this section we record the following observation about the maximum modul... |

1 |
Cyclotomic integers and finite geometry polynomials with ±1 coefficients
- Schmidt
- 1999
(Show Context)
Citation Context ...mials and the proof of the nonexistence of self-reciprocal Barker polynomials. In [23] and [24] Turyn and Storer showed that no even degree Barker polynomials exist for n > 12 (and indeed, as Schmidt =-=[21]-=- shows, none exist for any degree between 12 and 10 20 . It can also be shown (see Turyn 14[25]) that any odd degree Barker polynomial of degree greater than 12 must have degree of the form 4s 2 − 1,... |