Cassels Bases

  • Melvyn B. NathansonEmail author


This paper describes several classical constructions of thin bases of finite order in additive number theory, and, in particular, gives a complete presentation of a beautiful construction of Cassels of a class of polynomially asymptotic bases. Some open problems are also discussed.


Additive basis Sumset Thin basis Polynomially asymptotic basis Cassels basis Raikov-Stöhr basis Jia-Nathanson basis Additive number theory 


  1. 1.
    V. Blomer, Thin bases of order h, J. Number Theory 98 (2003), 34–46.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    J. W. S. Cassels, Über Basen der natürlichen Zahlenreihe, Abhandlungen Math. Seminar. Univ. Hamburg 21 (1975), 247–257.MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Chatrovsky, Sur les bases minimales de la suite des nombres naturels, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 335–340.Google Scholar
  4. 4.
    S. L. G. Choi, P. Erdős, and M. B. Nathanson, Lagrange’s theorem with N 1∕3 squares, Proc. Am. Math. Soc. 79 (1980), 203–205.zbMATHGoogle Scholar
  5. 5.
    P. Erdős and M. B. Nathanson, Lagrange’s theorem and thin subsequences of squares, Contributions to probability, Academic Press, New York, 1981, pp. 3–9.Google Scholar
  6. 6.
    G. Grekos, L. Haddad, C. Helou, and J. Pihko, Variations on a theme of Cassels for additive bases, Int. J. Number Theory 2 (2006), no. 2, 249–265.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Hofmeister, Thin bases of order two, J. Number Theory 86 (2001), no. 1, 118–132.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    X.-D. Jia, Minimal bases and g-adic representations of integers, Number Theory: New York Seminar 1991-1995 (New York), Springer-Verlag, New York, 1996, pp. 201–209.CrossRefGoogle Scholar
  9. 9.
    X.-D. Jia and M. B. Nathanson, A simple construction of minimal asymptotic bases, Acta Arith. 52 (1989), no. 2, 95–101.MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. B. Nathanson, Waring’s problem for sets of density zero, Analytic number theory (Philadelphia, PA, 1980), Lecture Notes in Math., vol. 899, Springer, Berlin, 1981, pp. 301–310.Google Scholar
  11. 11.
    M. B. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996.Google Scholar
  12. 12.
    M. B. Nathanson, Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory, J. Number Theory 129 (2009), 1608–1621.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    D. Raikov, Über die Basen der natürlichen Zahlentreihe, Mat. Sbornik N. S. 2 44 (1937), 595–597.Google Scholar
  14. 14.
    C. Schmitt, Uniformly thin bases of order two, Acta Arith. 124 (2006), no. 1, 17–26.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. Stöhr, Eine Basis h-Ordnung für die Menge aller natürlichen Zahlen, Math. Zeit. 42 (1937), 739–743.CrossRefGoogle Scholar
  16. 16.
    A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I, II, J. Reine Angew. Math. 194 (1955), 40–65, 111–140.Google Scholar
  17. 17.
    Van H. Vu, On a refinement of Waring’s problem, Duke Math. J. 105 (2000), no. 1, 107–134.Google Scholar
  18. 18.
    E. Wirsing, Thin subbases, Analysis 6 (1986), 285–308.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsLehman College (CUNY)BronxUSA
  2. 2.CUNY Graduate CenterNew YorkUSA

Personalised recommendations