## Fourier bases and a distance problem of Erdős (1999)

Venue: | Amer. J. Math |

Citations: | 10 - 2 self |

### BibTeX

@ARTICLE{Iosevich99fourierbases,

author = {Alex Iosevich and Nets Katz and Steen Pedersen},

title = {Fourier bases and a distance problem of Erdős},

journal = {Amer. J. Math},

year = {1999},

pages = {115--120}

}

### OpenURL

### Abstract

Abstract. We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases. Let D be a domain in R d, i.e., D is a Lebesgue measurable subset of R d with finite non-zero Lebesgue measure. We say that D is a spectral set if L 2 (D) has orthogonal basis of the form EΛ = {e 2πix·λ}

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Citation Context ...tly, Chung, Szeremedi, and Trotter proved that g2(n) ≥ C n 4 5 log c (n) for some c > 0. See [CST]. Theorem 2 above is proved by induction using the g2(n) ≥ Cn 3 4 result proved by Clarkson et al. in =-=[C]-=-. As the reader shall see, Theorem 1 does not require the full strength of Theorem 2. We just need the fact gd(n) ≥ Cdn 1 d +ǫ, for some ǫ > 0. It is interesting to contrast the case of the ball with ... |

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Citation Context ...um number of distances determined by n points in R d . Then (*) gd(n) ≥ Cdn 3 3d−2 . Remark. The study of the problem addressed in Theorem 2 was initiated by Erdős. He proved that g2(n) ≥ Cn 1 2. See =-=[Erd]-=-. Moser proved in [Mos] that g2(n) ≥ Cn 2 3 . More recently, Chung, Szeremedi, and Trotter proved that g2(n) ≥ C n 4 5 log c (n) for some c > 0. See [CST]. Theorem 2 above is proved by induction using... |

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Citation Context ... the sense defined above, where D is a bounded domain. There exists an r > 0 so that any ball of radius r contains at least one point from Λ. Proof. This is a special case of [IoPe2]. See also [Beu], =-=[Lan]-=-, and [GrRa]. It is a consequence of Theorem 3 that if D is a spectral set then there exists a constant C > 0 such that if Λ is a spectrum for D then #{Λ ∩ Bd(R)} ≥ C R d for any ball Bd(R)4 ALEX IOS... |

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Citation Context ...rum for D. We say that a family D + t, t ∈ T, of translates of a domain D tiles R d if ∪t∈T (D + t) is a partition of R d up to sets of Lebesgue measure zero. Conjecture. It has been conjectured (see =-=[Fug]-=-) that a domain D is a spectral set if and only if it is possible to tile R d by a family of translates of D. This conjecture is nowhere near resolution, even in dimension one. It has been the subject... |

32 |
on Landau’s necessary density conditions for sampling and interpolation of band-limited functions
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Citation Context ...defined above, where D is a bounded domain. There exists an r > 0 so that any ball of radius r contains at least one point from Λ. Proof. This is a special case of [IoPe2]. See also [Beu], [Lan], and =-=[GrRa]-=-. It is a consequence of Theorem 3 that if D is a spectral set then there exists a constant C > 0 such that if Λ is a spectrum for D then #{Λ ∩ Bd(R)} ≥ C R d for any ball Bd(R)4 ALEX IOSEVICH, NETS ... |

30 | Spectral sets and factorizations of finite abelian groups
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Citation Context ...y if it is possible to tile R d by a family of translates of D. This conjecture is nowhere near resolution, even in dimension one. It has been the subject of recent research, see for example [JoPe2], =-=[LaWa]-=-, and [Ped]. In this paper we address the following special case of the conjecture. Let Bd = {x ∈ R d : |x| ≤ 1} denote the unit ball. We prove that Theorem 1. An affine image of D = Bd, d ≥ 2, is not... |

25 | Spectral Pairs in Cartesian Coordinates - Jorgensen, Pedersen - 1999 |

22 | Spectral and tiling properties of the unit cube
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Citation Context ...uire the full strength of Theorem 2. We just need the fact gd(n) ≥ Cdn 1 d +ǫ, for some ǫ > 0. It is interesting to contrast the case of the ball with the case of the cube [0, 1] d . It was proved in =-=[IoPe1]-=-, (and, independently, in [LRW]; for d ≤ 3 this was established in [JoPe2]), that Λ is a spectrum for [0, 1] d , in the sense defined above, if and only if Λ is a tiling set for [0, 1] d , in the sens... |

22 | Non-symmetric convex domains have no basis of exponentials
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Citation Context ...n the other hand, we will show that the number of distinct distances between the elements of {Λ ∩ Bd(R)} is ≈ R. Theorem 2 implies that if R is sufficiently large, this is not possible. Kolountzakis (=-=[Kol]-=-) recently proved that if D is any convex non-symmetric domain in Rd , then D is not a spectral set. Theorem 1 is a step in the direction of proving that if D is a convex domain such that ∂D has at le... |

18 | Orthonormal bases of exponentials for the n-cube
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(Show Context)
Citation Context ... 2. We just need the fact gd(n) ≥ Cdn 1 d +ǫ, for some ǫ > 0. It is interesting to contrast the case of the ball with the case of the cube [0, 1] d . It was proved in [IoPe1], (and, independently, in =-=[LRW]-=-; for d ≤ 3 this was established in [JoPe2]), that Λ is a spectrum for [0, 1] d , in the sense defined above, if and only if Λ is a tiling set for [0, 1] d , in the sense that [0, 1] d + Λ = Rd withou... |

14 |
Local harmonic analysis with some applications to differential operators, Some recent advances in the basic sciences, Academic press 1
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Citation Context ...or D in the sense defined above, where D is a bounded domain. There exists an r > 0 so that any ball of radius r contains at least one point from Λ. Proof. This is a special case of [IoPe2]. See also =-=[Beu]-=-, [Lan], and [GrRa]. It is a consequence of Theorem 3 that if D is a spectral set then there exists a constant C > 0 such that if Λ is a spectrum for D then #{Λ ∩ Bd(R)} ≥ C R d for any ball Bd(R)4 A... |

11 | Average decay of Fourier transforms and integer points in polyhedra, Ark - Brandolini, Colzani, et al. - 1997 |

9 | Orthogonal harmonic analysis of fractal measures
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Citation Context ...f and only if it is possible to tile R d by a family of translates of D. This conjecture is nowhere near resolution, even in dimension one. It has been the subject of recent research, see for example =-=[JoPe2]-=-, [LaWa], and [Ped]. In this paper we address the following special case of the conjecture. Let Bd = {x ∈ R d : |x| ≤ 1} denote the unit ball. We prove that Theorem 1. An affine image of D = Bd, d ≥ 2... |

8 | How wide are the spectral gaps
- Iosevich, Pedersen
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(Show Context)
Citation Context ... Λ is a spectrum for D in the sense defined above, where D is a bounded domain. There exists an r > 0 so that any ball of radius r contains at least one point from Λ. Proof. This is a special case of =-=[IoPe2]-=-. See also [Beu], [Lan], and [GrRa]. It is a consequence of Theorem 3 that if D is a spectral set then there exists a constant C > 0 such that if Λ is a spectrum for D then #{Λ ∩ Bd(R)} ≥ C R d for an... |

6 |
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(Show Context)
Citation Context ... by Erdős. He proved that g2(n) ≥ Cn 1 2. See [Erd]. Moser proved in [Mos] that g2(n) ≥ Cn 2 3 . More recently, Chung, Szeremedi, and Trotter proved that g2(n) ≥ C n 4 5 log c (n) for some c > 0. See =-=[CST]-=-. Theorem 2 above is proved by induction using the g2(n) ≥ Cn 3 4 result proved by Clarkson et al. in [C]. As the reader shall see, Theorem 1 does not require the full strength of Theorem 2. We just n... |

4 |
Keller’s cube tiling conjecture is false in high dimensions
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Citation Context ...0, 1] d + Λ = Rd without overlaps. It follows that [0, 1] d has lots of spectra. The standard integer lattice Λ = Zd is an example, though there are many non-trivial examples as well. See [IoPe1] and =-=[LaSh]-=-. Our method of proof is as follows. We shall argue that if Bd were a spectral set, then any corresponding spectrum Λ would have the property #{Λ ∩ Bd(R)} ≈ Rd , where Bd(R) denotes a ball of radius R... |

1 |
On different distances determined by n points, American Mathematical Monthly 59
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Citation Context ... self-adjoint operators A and B acting on the same Hilbert space commute if the bounded unitary operators exp ( − √ −1sA ) and exp ( − √ −1tB ) commute for all real numbers s and t. See, for example, =-=[ReSi]-=- for more details on the needed operator theory. As an immediate consequence of [Fug] and Theorem 1 we have: 1991 Mathematics Subject Classification. 42B. Research supported in part by NSF grants DMS9... |