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Number Theory pp 123-139 | Cite as

Additive problems in combinatorial number theory

  • Melvyn B. Nathanson
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1383)

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References

  1. 1.
    J. W. S. Cassels, Über Basen der natürlichen Zahlenreihe, Abh. Math. Sem. Univ. Hamburg 21 (1957), 247–257.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L. Chatrovsky, Sur les bases minimales de la suite des nombres naturels (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 335–340.MathSciNetGoogle Scholar
  3. 3.
    S. L. G. Choi, P. Erdös, and M. B. Nathanson, Lagrange's theorem with N1/3 squares, Proc. of the Amer. Math. Soc. 79 (1980), 203–205.zbMATHGoogle Scholar
  4. 4.
    P. Erdös, Einige Bemerkungen zur Arbeit von A. Stöhr "Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe," J. reine angew. Math. 197 (1957), 216–219.MathSciNetGoogle Scholar
  5. 5.
    P. Erdös, On the multiplicative representations of integers, Israel J. Math. 2 (1964), 251–261.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    P. Erdös and R. Freud, personal communication.Google Scholar
  7. 7.
    P. Erdös and E. Härtter, Konstruktion von nichtperiodischen Minimalbasen mit der dichte 1/2 für die Menge der nichtnegativen ganzen Zahlen, J. reine angew. Math. 221 (1966), 44–47.MathSciNetGoogle Scholar
  8. 8.
    P. Erdös and M. B. Nathanson, Oscillations of bases for the natural numbers, Proc. Amer. Math. Soc. 53 (1975), 253–258.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Erdös and M. B. Nathanson, Systems of distinct representatives and minimal bases in additive number theory, in: M. B. Nathanson, ed., Number Theory, Carbondale 1979, Lecture Notes in Mathematics, vol. 751, Springer-Verlag, Berlin-New York, 1979, pp. 89–107.CrossRefGoogle Scholar
  10. 10.
    P. Erdös and M. B. Nathanson, Lagrange's theorem and thin subsequences of squares, in: J. Gani and V. K. Rohatgi (eds.), Contributions to Probability, Academic Press, New York, 1981, pp. 3–9.CrossRefGoogle Scholar
  11. 11.
    P. Erdös and M. B. Nathanson, Minimal asymptotic bases with prescribed densities, Illinois J. Math. 32 (1988), 562–574.MathSciNetzbMATHGoogle Scholar
  12. 12.
    P. Erdös and M. B. Nathanson, Asymptotic bases with many representations, Acta Arith., to appear.Google Scholar
  13. 13.
    P. Erdös, M. B. Nathanson, and A. Sárközy, Sumsets containing infinite arithmetic progressions, J. Number Theory 28 (1988), 159–166.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. Erdös and P. Turán, On a problem of Sidon in additive number theory and some related questions, J. London Math. Soc. 16 (1941), 212–215.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Filaseta, Sets with elements summing to square-free numbers, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), 243–246.MathSciNetzbMATHGoogle Scholar
  16. 16.
    G. Freiman, On two additive problems, to appear.Google Scholar
  17. 17.
    E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. reine angew. Math. 196 (1956), 170–204.MathSciNetzbMATHGoogle Scholar
  18. 18.
    X.-D. Jia and M. B. Nathanson, A simple construction of minimal asymptotic bases, Acta Arith. 52 (1988), to appear.Google Scholar
  19. 19.
    M. Kneser, Abschätzungen der asymptotischen Dichte von Summenmengen, Math. Zeit. 58 (1953), 459–484.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974), 324–333.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. B. Nathanson, Waring's problem for sets of density zero, in: M. I. Knopp (ed.), Number Theory, Philadelphia 1980, Lecture Notes in Mathematics, Vol. 899, Springer-Verlag, Berlin-New York, 1981, pp. 301–310.Google Scholar
  22. 22.
    M. B. Nathanson, Waring's problem for finite intervals, Proc. Amer. Math. Soc. 96 (1986), 15–17.MathSciNetzbMATHGoogle Scholar
  23. 23.
    M. B. Nathanson, A generalization of the Goldbach-Shnirel'man theorem, Amer. Math. Monthly 94 (1987), 768–771.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. of the Amer. Math. Soc. 99 (1987), 22–24.MathSciNetzbMATHGoogle Scholar
  25. 25.
    M. B. Nathanson, Multiplicative representations of integers, Israel J. Math. 57 (1987), 129–136.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. B. Nathanson, Minimal bases and powers of 2, Acta Arith. 49 (1988), 525–532.MathSciNetzbMATHGoogle Scholar
  27. 27.
    M. B. Nathanson, Sumsets containing k-free integers, Journées Arithmétiques de Ulm, 14–18 September 1987, to appear.Google Scholar
  28. 28.
    M. B. Nathanson and A. Sárközy, On the maximum density of minimal asymptotic bases, Proc. Amer. Math. Soc., to appear.Google Scholar
  29. 29.
    M. B. Nathanson and A. Sárközy, Sumsets containing long arithmetic progressions and powers of 2, Acta Arith., to appear.Google Scholar
  30. 30.
    J. Nešetřil and V. Rödl, Two proofs in combinatorial number theory, Proc. Amer. Math. Soc. 93 (1985), 185–188.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    D. Raikov, Über die Basen der natürlichen Zahlenreihe, Mat. Sbor. N.S. 2 44 (1937), 595–597.zbMATHGoogle Scholar
  32. 32.
    L. G. Shnirel'man, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933), 649–690.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Stöhr, Eine Basis h-ter Ordnung für die Menge aller natürlichen Zahlen, Math. Zeit. 42 (1937), 739–743.CrossRefzbMATHGoogle Scholar
  34. 34.
    A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe, II, J. reine angew. Math. 194 (1955), 111–140.zbMATHGoogle Scholar
  35. 35.
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.MathSciNetzbMATHGoogle Scholar
  36. 36.
    E. Wirsing, Thin subbases, Analysis 6 (1986), 285–308.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    J. Zöllner, Der Vier-Quadrate-Satz und ein Problem von Erdös und Nathanson, Dissertation, Johannes Gutenberg-Universität, Mainz, 1984.zbMATHGoogle Scholar
  38. 38.
    J. Zöllner, Über eine Vermutung von Choi, Erdös, und Nathanson, Acta Arith. 45 (1985), 211–213.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Melvyn B. Nathanson
    • 1
  1. 1.Office of the Provost and Vice President for Academic AffairsLehman College (CUNY)Bronx

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