Skip to main content

Addictive Number Theory

  • Chapter
  • First Online:
Additive Number Theory

Abstract

In 1996, just after Springer-Verlag published my books Additive Number Theory: The Classical Bases [34] and Additive Number Theory: Inverse Problems and the Geometry of Sumsets [35], I went into my local Barnes and Noble superstore on Route 22 in Springfield, New Jersey, and looked for them on the shelves. Suburban bookstores do not usually stock technical mathematical books, and, of course, the books were not there. As an experiment, I asked if they could be ordered. The person at the information desk typed in the titles, and told me that his computer search reported that the books did not exist. However, when I gave him the ISBN numbers, he did find them in the Barnes and Noble database. The problem was that the book titles had been cataloged incorrectly. The data entry person had written Addictive Number Theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    I have told this story many times, and like every good story, it has acquired an independent existence. I have heard others tell variations on the tale, always with the same additive–addictive punch line.

References

  1. J. Batson, Nathanson heights in finite vector spaces, J. Number Theory 128 (2008), no. 9, 2616–2633.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Borisov, M. B. Nathanson, and Y. Wang, Quantum integers and cyclotomy, J. Number Theory 109 (2004), no. 1, 120–135.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. W. S. Cassels, On the equation \({a}^{x} - {b}^{y} = 1\), Am. J. Math. 75 (1953), 159–162.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. W. S. Cassels, On the equation \({a}^{x} - {b}^{y} = 1\) . II, Proc. Cambridge Philos. Soc. 56 (1960), 97–103.

    Article  MathSciNet  Google Scholar 

  5. J. Cilleruelo and M. B. Nathanson, Dense sets of integers with prescribed representation functions, preprint, 2007.

    Google Scholar 

  6. P. Erdős and M. B. Nathanson, Maximal asymptotic nonbases, Proc. Am. Math. Soc. 48 (1975), 57–60.

    Google Scholar 

  7. P. Erdős and M. B. Nathanson, Partitions of the natural numbers into infinitely oscillating bases and nonbases, Comment. Math. Helv. 51 (1976), no. 2, 171–182.

    Article  MathSciNet  Google Scholar 

  8. P. Erdős and M. B. Nathanson, Systems of distinct representatives and minimal bases in additive number theory, Number theory, Carbondale 1979 (Proceedings of the Southern Illinois Conference, Southern Illinois University, Carbondale, Ill., 1979), Lecture Notes in Mathematics, vol. 751, Springer, Berlin, 1979, pp. 89–107.

    Google Scholar 

  9. P. Erdős and E. Szemerédi, On sums and products of integers, Studies in Pure Mathematics, To the Memory of Paul Turán (P. Erdős, L. Alpár, G. Halász, and A. Sárközy, eds.), Birkhäuser Verlag, Basel, 1983, pp. 213–218.

    Google Scholar 

  10. Harvard University Nuclear Nonproliferation Study Group, Nuclear nonproliferation: the spent fuel problem, Pergamon policy studies on energy and environment, Pergamon Press, New York, 1979.

    Google Scholar 

  11. H. Halberstam and K. F. Roth, Sequences, Vol. 1, Oxford University Press, Oxford, 1966, Reprinted by Springer-Verlag, Heidelberg, in 1983.

    MATH  Google Scholar 

  12. S.-P. Han, C. Kirfel, and M. B. Nathanson, Linear forms in finite sets of integers, Ramanujan J. 2 (1998), no. 1–2, 271–281.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math. 196 (1956), 170–204.

    MathSciNet  MATH  Google Scholar 

  14. X.-D. Jia and M. B. Nathanson, A simple construction of minimal asymptotic bases, Acta Arith. 52 (1989), no. 2, 95–101.

    MathSciNet  MATH  Google Scholar 

  15. A. G. Khovanskiĭ, The Newton polytope, the Hilbert polynomial and sums of finite sets, Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 57–63, 96.

    MathSciNet  Google Scholar 

  16. A. G. Khovanskiĭ, Sums of finite sets, orbits of commutative semigroups and Hilbert functions, Funktsional. Anal. i Prilozhen. 29 (1995), no. 2, 36–50, 95.

    MathSciNet  Google Scholar 

  17. A. V. Kontorovich and M. B. Nathanson, Quadratic addition rules for quantum integers, J. Number Theory 117 (2006), no. 1, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974, reprinted by Dover Publications in 2006.

    MATH  Google Scholar 

  19. E. Landau, Handbuch der Lehre von der verteilung der primzahlen, Chelsea Publishing Company, New York, 1909, reprinted by Chelsea in 1974.

    Google Scholar 

  20. J. Lee, Geometric structure of sumsets, preprint, 2007.

    Google Scholar 

  21. 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.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. B. Nathanson, Derivatives of binary sequences, SIAM J. Appl. Math. 21 (1971), 407–412.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. B. Nathanson, Complementing sets of n-tuples of integers, Proc. Am. Math. Soc. 34 (1972), 71–72.

    MathSciNet  MATH  Google Scholar 

  24. M. B. Nathanson, An exponential congruence of Mahler, Am. Math. Monthly 79 (1972), 55–57.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. B. Nathanson, Integrals of binary sequences, SIAM J. Appl. Math. 23 (1972), 84–86.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. B. Nathanson, On the greatest order of an element of the symmetric group, Am. Math. Monthly 79 (1972), 500–501.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. B. Nathanson, Sums of finite sets of integers, Am. Math. Monthly 79 (1972), 1010–1012.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. B. Nathanson, On the fundamental domain of a discrete group, Proc. Am. Math. Soc. 41 (1973), 629–630.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. B. Nathanson, Catalan’s equation in K(t), Am. Math. Monthly 81 (1974), 371–373.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974), 324–333.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. B. Nathanson, Multiplication rules for polynomials, Proc. Am. Math. Soc. 69 (1978), no. 2, 210–212. MR MR0466087 (57 #5970)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. B. Nathanson, Komar-Melamid: Two Soviet Dissident Artists, Southern Illinois University Press, Carbondale, IL, 1979.

    Google Scholar 

  33. M. B. Nathanson, Classification problems in K-categories, Fund. Math. 105 (1979/80), no. 3, 187–197.

    MathSciNet  Google Scholar 

  34. M. B. Nathanson, Additive number theory: the classical bases, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996.

    Google Scholar 

  35. M. B. Nathanson, Additive number theory: inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996.

    Google Scholar 

  36. M. B. Nathanson, On sums and products of integers, Proc. Am. Math. Soc. 125 (1997), no. 1, 9–16.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. B. Nathanson, Growth of sumsets in abelian semigroups, Semigroup Forum 61 (2000), no. 1, 149–153.

    Article  MathSciNet  MATH  Google Scholar 

  38. M. B. Nathanson, A functional equation arising from multiplication of quantum integers, J. Number Theory 103 (2003), no. 2, 214–233.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. B. Nathanson, Unique representation bases for the integers, Acta Arith. 108 (2003), no. 1, 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. B. Nathanson, Formal power series arising from multiplication of quantum integers, Unusual applications of number theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 64, American Mathematical Society, Providence, RI, 2004, pp. 145–167.

    Google Scholar 

  41. M. B. Nathanson, The inverse problem for representation functions of additive bases, Number theory (New York, 2003), Springer, New York, 2004, pp. 253–262.

    Google Scholar 

  42. M. B. Nathanson, Representation functions of additive bases for abelian semigroups, Int. J. Math. Math. Sci. (2004), no. 29–32, 1589–1597.

    Article  MathSciNet  Google Scholar 

  43. M. B. Nathanson, Every function is the representation function of an additive basis for the integers, Port. Math. (N.S.) 62 (2005), no. 1, 55–72.

    MathSciNet  MATH  Google Scholar 

  44. M. B. Nathanson, Heights on the finite projective line, Intern. J. Number Theory 5 (2009), 55–65.

    Article  MathSciNet  MATH  Google Scholar 

  45. M. B. Nathanson, Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory, preprint, 2009.

    Google Scholar 

  46. M. B. Nathanson, Phase transitions in infinitely generated groups, and a problem in additive number theory, Integers, to appear.

    Google Scholar 

  47. M. B. Nathanson, Nets in groups, minimum length g-adic representations, and minimal additive complements, preprint, 2009.

    Google Scholar 

  48. 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.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. B. Nathanson and I. Z. Ruzsa, Polynomial growth of sumsets in abelian semigroups, J. Théor. Nombres Bordeaux 14 (2002), no. 2, 553–560.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. B. Nathanson and B. D. Sullivan, Heights in finite projective space, and a problem on directed graphs, Integers 8 (2008), A13, 9.

    MathSciNet  Google Scholar 

  51. K. O’Bryant, Gaps in the spectrum of Nathanson heights of projective points, Integers 7 (2007), A38, 7 pp. (electronic).

    Google Scholar 

  52. S. L. Segal, On Nathanson’s functional equation, Aequationes Math. 28 (1985), no. 1–2, 114–123.

    Article  MathSciNet  MATH  Google Scholar 

  53. S. L. Segal, Mathematicians under the Nazis, Princeton University Press, Princeton, NJ, 2003.

    MATH  Google Scholar 

  54. S. L. Segal, Nine introductions in complex analysis, revised ed., North-Holland Mathematics Studies, vol. 208, Elsevier Science B.V., Amsterdam, 2008.

    MATH  Google Scholar 

  55. A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I, II, J. Reine Angew. Math. 194 (1955), 40–65, 111–140.

    MathSciNet  MATH  Google Scholar 

  56. S. S. Wagstaff, Solution of Nathanson’s exponential congruence, Math. Comp. 33 (1979), 1097–1100.

    MathSciNet  MATH  Google Scholar 

  57. A. Weil, Elliptic Functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin, 1976.

    Book  MATH  Google Scholar 

Download references

Acknowledgements

I want to thank David and Gregory Chudnovsky for organizing and editing this volume. Back in 1982, the Chudnovskys and I, together with Harvey Cohn, created the New York Number Theory Seminar at the CUNY Graduate Center, and we have been running this weekly seminar together for more than a quarter century. It has been a pleasure to know them and work with them.

Most of all, I acknowledge the love and support of my wife Marjorie and children Becky and Alex.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Melvyn B. Nathanson .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Nathanson, M.B. (2010). Addictive Number Theory. In: Chudnovsky, D., Chudnovsky, G. (eds) Additive Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68361-4_1

Download citation

Publish with us

Policies and ethics