### Abstract

Let A ⊆ {1,..., N} and let {a _{n}} _{n∈Λ} be a sequence with \a _{n}\ ≤ 1 for all n. It is easy to see that || _{n∈Λ}∑a _{n}e(nθ)|| _{p}≤|| _{n∈Λ}∑e(nθ)|| _{p} for every even integer p. We give an example which shows that this statement can fail rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

Original language | English |
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Pages (from-to) | 511-517 |

Number of pages | 7 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 137 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 2004 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On the hardy-littlewood majorant problem.** / Green, Ben; Ruzsa, I.

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 137, no. 3, pp. 511-517. https://doi.org/10.1017/S0305004104007911

}

TY - JOUR

T1 - On the hardy-littlewood majorant problem

AU - Green, Ben

AU - Ruzsa, I.

PY - 2004/11

Y1 - 2004/11

N2 - Let A ⊆ {1,..., N} and let {a n} n∈Λ be a sequence with \a n\ ≤ 1 for all n. It is easy to see that || n∈Λ∑a ne(nθ)|| p≤|| n∈Λ∑e(nθ)|| p for every even integer p. We give an example which shows that this statement can fail rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

AB - Let A ⊆ {1,..., N} and let {a n} n∈Λ be a sequence with \a n\ ≤ 1 for all n. It is easy to see that || n∈Λ∑a ne(nθ)|| p≤|| n∈Λ∑e(nθ)|| p for every even integer p. We give an example which shows that this statement can fail rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

UR - http://www.scopus.com/inward/record.url?scp=10044269728&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10044269728&partnerID=8YFLogxK

U2 - 10.1017/S0305004104007911

DO - 10.1017/S0305004104007911

M3 - Article

AN - SCOPUS:10044269728

VL - 137

SP - 511

EP - 517

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -