Problems in additive number theory, III

  • Melvyn B. Nathanson
Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)


Let ℕ, ℕ0, ℤ and ℕ d denote, respectively, the sets of positive integers, non-negative integers, integers and d-dimensional integral lattice points. Let G denote an arbitrary abelian group and let X denote an arbitrary abelian semigroup, written additively. Let |S| denote the cardinality of the set S. For any sets A and B, we write A∼B if their symmetric difference is finite, that is, if |(A \ B) ∪ (B \ A) | < ∞.


Linear Form Representation Function Cayley Graph Arithmetic Progression Counting Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Biró, Divisibility of integer polynomials and tilings of the integers, Acta Arith. 118 (2005), no. 2, 117–127.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    B. Bukh, Sums of dilates, arXiv preprint 0711.1610, 2007.Google Scholar
  3. [3]
    J. W. S. Cassels, Über Basen der natürlichen Zahlenreihe, Abh. Math. Sem. Univ. Hamburg 21 (1975), 247–257.CrossRefMathSciNetGoogle Scholar
  4. [4]
    G. A. Dirac, Note on a problem in additive number theory, J. London Math. Soc. 26 (1951), 312–313.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Erdős and M. B. Nathanson, Oscillations of bases for the natural numbers, Proc. Amer. Math. Soc. 53 (1975), no. 2, 253–258.CrossRefMathSciNetGoogle Scholar
  6. [6]
    -, Partitions of the natural numbers into infinitely oscillating bases and non-bases, Comment. Math. Helv. 51 (1976), no. 2, 171–182.CrossRefMathSciNetGoogle Scholar
  7. [7]
    -, Systems of distinct representatives and minimal bases in additive number theory, Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979.), Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 89–107.CrossRefGoogle Scholar
  8. [8]
    P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Y. O. Hamidoune, An application of connectivity theory in graphs to factorizations of elements in groups, European J. Combin. 2 (1981), no. 4, 349–355.MATHMathSciNetGoogle Scholar
  10. [10]
    -, A note on minimal directed graphs with given girth, J. Combin. Theory Ser. B 43 (1987), no. 3, 343–348.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math. 196 (1956), 170–204.MATHMathSciNetGoogle Scholar
  12. [12]
    P. V. Hegarty, Some explicit constructions of sets with more sums than differences, Acta Arith. 130 (2007), 61–77.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    P. V. Hegarty and S. J. Miller, When almost all sets are difference dominated, arXiv preprint 0707.3417, 2007.Google Scholar
  14. [14]
    J. Hennefeld, Asymptotic non-bases which are not subsets of maximal aymptotic non-bases, Proc. Amer. Math. Soc. 62 (1977), 23–24.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    C. Vinuesa, J. Cilleruelo and M. Silva, A sumset problem, preprint, 2008.Google Scholar
  16. [16]
    J. H. B. Kemperman, On complexes in a semigroup, Indag. Math. 18 (1956), 247–254.MathSciNetGoogle Scholar
  17. [17]
    M. N. Kolountzakis, Translational tilings of the integers with long periods, Electron. J. Combin. 10 (2003), Research Paper 22, 9 pp.Google Scholar
  18. [18]
    Z. Ljujic and M. B. Nathanson, Complementing sets of integers with respect to a multiset, preprint, 2008.Google Scholar
  19. [19]
    G. Martin and K. O’Brvant, Many sets have more sums than differences, Additive Combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 287–305.Google Scholar
  20. [20]
    J. C. M. Nash and M. B. Nathanson, Cofinite subsets of asymptotic bases for the positive integers, J. Number Theory 20 (1985), no. 3, 363–372.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    M. B. Nathanson, Sums of finite sets of integers, Amer. Math. Monthly 79 (1972), 1010–1012.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    -, Minimal bases and maximal non-bases in additive number theory, J. Number Theory 6 (1974), 324–333.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    -, s-maximal non-bases of density zero, J. London Math. Soc. (2) 15 (1977), no. 1, 29–34. MR MR0435021 (55 #7983)MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    -, Unique representation bases for the integers, Acta Arith. 108 (2003), no. 1, 1–8.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    -, The inverse problem for representation functions of additive bases, Number Theory (New York, 2003), Springer, New York, 2004, pp. 253–262.Google Scholar
  26. [26]
    -, Every function is the representation function of an additive basis for the integers, Port. Math. (N.S.) 62 (2005), no. 1, 55–72.MATHMathSciNetGoogle Scholar
  27. [27]
    -, The Caccetta-Häggkvist conjecture and additive number theory, arXiv: math.CO/0603469, 2006.Google Scholar
  28. [28]
    -, Sets with more sums than differences, Integers 7 (2007), paper A5, 24 pp.Google Scholar
  29. [29]
    -, Inverse problems for linear forms over finite sets of integers, J. Ramanujan Math. Soc. 23 (2008), no. 2, 1–15.MathSciNetGoogle Scholar
  30. [30]
    -, Problems in additive number theory, I, Additive Combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 263–270.Google Scholar
  31. [31]
    -, Problems in additive number theory, II: Linear forms and complementing sets of integers, J. Théor. Nombres Bordeaux, to appear.Google Scholar
  32. [32]
    -, Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory, arXiv preprint 0806.0984, 2008.Google Scholar
  33. [33]
    M. B. Nathanson, K. O’Bryant, B. Orosz, I. Ruzsa, and M. Silva, Binary linear forms over finite sets of integers, Acta Arith. 129 (2007), 341–361.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    D. J. Newman, Tesselation of integers, J. Number Theory 9 (1977), no. 1, 107–111.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    I. Z. Ruzsa, Appendix in R. Tijdeman, “Periodicity and almost-periodicity”, 2006.Google Scholar
  36. [36]
    R. Tijdeman, Periodicity and Almost-Periodicity, More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin, 2006, pp. 381–405.Google Scholar

Copyright information

© Birkhäuser Verlag 2009

Authors and Affiliations

  • Melvyn B. Nathanson
    • 1
    • 2
  1. 1.Department of MathematicsLehman College (CUNY)BronxUSA
  2. 2.CUNY Graduate CenterNew YorkUSA

Personalised recommendations