Bohr Cluster Points of Sidon Sets

L. Ramsey

Colloquium Mathematicae (1995)

  • Volume: 68, Issue: 2, page 285-290
  • ISSN: 0010-1354

Abstract

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It is a long standing open problem whether Sidon subsets of ℤ can be dense in the Bohr compactification of ℤ ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of ℤ, one can construct from it another Sidon set which is dense in the Bohr compactification of ℤ. A weaker result holds for quasi-independent and dissociate subsets of ℤ.

How to cite

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Ramsey, L.. "Bohr Cluster Points of Sidon Sets." Colloquium Mathematicae 68.2 (1995): 285-290. <http://eudml.org/doc/210312>.

@article{Ramsey1995,
abstract = {It is a long standing open problem whether Sidon subsets of ℤ can be dense in the Bohr compactification of ℤ ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of ℤ, one can construct from it another Sidon set which is dense in the Bohr compactification of ℤ. A weaker result holds for quasi-independent and dissociate subsets of ℤ.},
author = {Ramsey, L.},
journal = {Colloquium Mathematicae},
keywords = {Bohr compactification; Sidon; quasi-independent; dissociate; Bohr cluster points; Sidon subsets; random process},
language = {eng},
number = {2},
pages = {285-290},
title = {Bohr Cluster Points of Sidon Sets},
url = {http://eudml.org/doc/210312},
volume = {68},
year = {1995},
}

TY - JOUR
AU - Ramsey, L.
TI - Bohr Cluster Points of Sidon Sets
JO - Colloquium Mathematicae
PY - 1995
VL - 68
IS - 2
SP - 285
EP - 290
AB - It is a long standing open problem whether Sidon subsets of ℤ can be dense in the Bohr compactification of ℤ ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of ℤ, one can construct from it another Sidon set which is dense in the Bohr compactification of ℤ. A weaker result holds for quasi-independent and dissociate subsets of ℤ.
LA - eng
KW - Bohr compactification; Sidon; quasi-independent; dissociate; Bohr cluster points; Sidon subsets; random process
UR - http://eudml.org/doc/210312
ER -

References

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  1. [K] Y. Katznelson, Sequences of integers dense in the Bohr group, in: Proc. Roy. Inst. Techn., June 1973, 73-86. 
  2. [LR] J. M. López and K. A. Ross, Sidon Sets, Marcel Dekker, New York, 1975, pp. 19-52. 
  3. [P] G. Pisier, Arithmetic characterization of Sidon sets, Bull. Amer. Math. Soc. 8 (1983), 87-89. Zbl0505.43002

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