Rocky Mountain Journal of Mathematics

Characterizations of Classes of $I_0$ Sets in Discrete Abelian Groups

Colin C. Graham and Kathryn E. Hare

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 40, Number 2 (2010), 513-525.

Dates
First available in Project Euclid: 13 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1273783903

Digital Object Identifier
doi:10.1216/RMJ-2010-40-2-513

Mathematical Reviews number (MathSciNet)
MR2646455

Zentralblatt MATH identifier
1201.42004

Subjects
Primary: 42A55: Lacunary series of trigonometric and other functions; Riesz products 42A63: Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
Secondary: 43A05: Measures on groups and semigroups, etc. 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Keywords
Associated sets Bohr group $\epsilon$-Kronecker sets FatouZygmund property $\epsilon$-free setsr Hadamard sets $I_0$ sets Sidon sets

Citation

Graham, Colin C.; Hare, Kathryn E. Characterizations of Classes of $I_0$ Sets in Discrete Abelian Groups. Rocky Mountain J. Math. 40 (2010), no. 2, 513--525. doi:10.1216/RMJ-2010-40-2-513. https://projecteuclid.org/euclid.rmjm/1273783903


Export citation

References

  • M. Déchamps-Gondim, Ensembles de Sidon topologiques, Ann. Inst. Fourier (Grenoble) 22 (1972), 51-79.
  • E.K. van Douwen, The maximal totally bounded group topology on $G$ and the biggest minimal $G$-space, for abelian groups $G$, Topology Appl. 34 (1990), 69-91.
  • J. Galindo and S. Hernandez, The concept of boundedness and the Bohr compactification of a MAP abelian group, Fund. Math. 159 (1999), 195-218.
  • B.N. Givens and K. Kunen, Chromatic numbers and Bohr topologies, Topology Appl. 131 (2003), 189-202.
  • C.C. Graham and K.E. Hare, $\varepsilon $ -Kronecker and $I_0$ sets in abelian groups, I: Arithmetic properties of $\varepsilon $-Kronecker sets, Math. Proc. Camb. Phil. Soc. 140 (2006), 475-489.
  • --------, $\varepsilon$-Kronecker and $I_0$ sets in abelian groups, III: Interpolation by measures on small sets, Stud. Math. 171 (2005), 15-32.
  • --------, $\varepsilon$-Kronecker and $I_0$ sets in abelian groups, IV: Interpolation by non-negative measures, Stud. Math. 177 (2006), 9-24.
  • --------, Characterizing Sidon sets by interpolation properties of subsets, Colloq. Math. 112 (2008), 175-199.
  • C.C. Graham, K.E. Hare and T.W. Körner, $\varepsilon$-Kronecker and $I_0$ sets in abelian groups, II: Sparseness of products of $\varepsilon $-Kronecker sets, Math. Proc. Camb. Phil. Soc. 140 (2006), 491-508.
  • C.C. Graham and A.T.M. Lau, Relative weak compactness of orbits in Banach spaces associated with locally compact groups, Trans. Amer. Math. Soc. 359 (2007), 1129-1160.
  • C.C. Graham and O.C. McGehee, Essays in commutative harmonic analysis, Springer-Verlag, New York, 1979.
  • K.E. Hare and L.T. Ramsey, $I_0$ sets in non-abelian groups, Math. Proc. Camb. Phil. Soc. 135 (2003), 81-98.
  • S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, Colloq. Math. 12 (1964), 23-39.
  • J.-P. Kahane, Ensembles de Ryll-Nardzewski et ensembles de Helson, Colloq. Math. 15 (1966), 87-92.
  • --------, Séries de Fourier Absolument Convergentes, Springer, Berlin, 1970
  • K. Kunen and W. Rudin, Lacunarity and the Bohr topology, Math. Proc. Camb. Phil. Soc. 126 (1999), 117-137.
  • J. Lopez and K. Ross, Sidon sets, Lecture Notes Pure Appl. Math. 13, Marcel Dekker, New York, 1975.
  • J.-F. Méla, Sur les ensembles d'interpolation de C. Ryll-Nardzewski et de S. Hartman, Studia Math. 29 (1968), 167-193.
  • L.T. Ramsey, A theorem of Ryll-Nardzewski and metrizable L.C.A. groups, Proc. Amer. Math. Soc. 78 (1980), 221-224.
  • --------, Comparisons of Sidon and $I_0$ sets, Colloq. Math. 70 (1996), 103-132.
  • B.P. Smith, Helson sets not containing the identity are uniform Fatou-Zygnund sets, Indiana Univ. Math. J. 27 (1978), 331-347.
  • E. Strzelecki, On a problem of interpolation by periodic and almost periodic functions, Colloq. Math. 11 (1963), 91-99.