Melvyn B.

Nathanson: Books
Monographs
1. Additive Number Theory: The Classical Bases, Graduate Texts in Mathe-
matics, Vol. 164, Springer-Verlag, New York, 1996.
2. Additive Number Theory: Inverse Problems and the Geometry of Sumsets,
Graduate Texts in Mathematics, Vol. 165, Springer-Verlag, New York,
1996.
3. Elementary Methods in Number Theory, Graduate Texts in Mathematics,
Vol. 195, Springer-Verlag, New York, 2000.
4. Additive Number Theory: Extremal Problems and the Combinatorics of
Sumsets, Graduate Texts in Mathematics, Springer, New York, 2009, to
appear.
5. Additive Number Theory: Density Problems and the Growth of Sumsets,
Graduate Texts in Mathematics, Springer, New York, 2010, to appear.
Proceedings
6. Number Theory Day: Proceedings of the Conference held at Rockefeller
University, New York, March 4, 1976, Edited by M. B. Nathanson, Lecture
Notes in Mathematics, Vol. 626, Springer-Verlag, Berlin, 1977.
7. Number Theory, Carbondale 1979: Proceedings of the Southern Illinois
Number Theory Conference, held at Southern Illinois University, Carbon-
dale, Ill., March 30–31, 1979, Edited by M. B. Nathanson, Lecture Notes
in Mathematics, Vol. 751, Springer, Berlin, 1979.
8. Number Theory: Proceedings of the seminar held at the City University of
New York, New York, 1982, Edited by D. V. Chudnovsky, G. V. Chud-
novsky, H. Cohn, and M. B. Nathanson, Lecture Notes in Mathematics,
Vol. 1052, Springer-Verlag, Berlin, 1984.
9. Number Theory: Proceedings of the seminar held at the City University
of New York, New York, 1983–1984, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, H. Cohn, and M. B. Nathanson, Lecture Notes in Mathemat-
ics, Vol. 1135, Springer-Verlag, Berlin, 1985.
10. Number Theory: Proceedings of the seminar held at the City University
of New York, New York, 1984–1985, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, H. Cohn, and M. B. Nathanson, Lecture Notes in Mathemat-
ics, Vol. 1240, Springer-Verlag, Berlin, 1987.
11. Number Theory: Proceedings of the seminar held at the City University
of New York, New York, 1985–1988, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, H. Cohn, and M. B. Nathanson, Lecture Notes in Mathemat-
ics, Vol. 1383, Springer-Verlag, Berlin, 1989.
12. Number Theory: Papers from the seminar held at the City University of
New York, New York, 1989–1990, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, H. Cohn, and M. B. Nathanson, Springer-Verlag, New York,
1991
13. Number Theory: Papers from the seminars held at the City University of
New York, New York, 1991–1995, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, and M. B. Nathanson, Springer-Verlag, New York, 1996.
14. Number Theory: Papers from the seminar (NYNTS) held at the City Uni-
versity of New York, New York, 2003, Edited by D. V. Chudnovsky, G. V.
Chudnovsky, and M. B. Nathanson, Springer-Verlag, New York, 2004.
1
2

15. Unusual Applications of Number Theory: Proceedings of the DIMACS Work-

shop held at Rutgers University, January 10–14, 2000, Edited by M.B.
Nathanson, DIMACS Series in Discrete Mathematics and Theoretical Com-
puter Science, Vol. 64, American Mathematical Society, Providence, RI,
2004.
16. Combinatorial Number Theory, Edited by B. M. Landman, M. B. Nathanson,
J. Nešetřil, and C. Pomerance, de Gruyter, Berlin, 2007
17. Additive Combinatorics, Edited by A. Granville, M. B. Nathanson, and J.
Solymosi, Amer. Math. Soc., Providence, 2007.
Translations
18. Anatolij A. Karatsuba, Basic Analytic Number Theory, Translated from the
second (1983) Russian Edition and with a preface by Melvyn B. Nathanson,
Springer-Verlag, Berlin, 1993.
19. Grigori Freiman, It Seems I am a Jew: A Samizdat Essay on Soviet Mathe-
matics, Translated from the Russian and with an introduction by Melvyn B.
Nathanson and appendices by Melvyn B. Nathanson and Andrei Sakharov,
Southern Illinois University Press, Carbondale, 1979
Other
20. Komar/Melamid: Two Soviet Dissident Artists, Southern Illinois Univer-
sity Press, Carbondale, 1979.
21. Nuclear Nonproliferation: The Spent Fuel Problem (Pergamon policy stud-
ies on energy and environment), Harvard University Nuclear Nonprolifera-
tion Study Group, Pergamon Press, 1979.

Melvyn B. Nathanson: Papers

1971
1. Derivatives of binary sequences, SIAM J. Appl. Math. 21 (1971), 407–412.
1972
2. An exponential congruence of Mahler, Amer. Math Monthly 79 (1972),
55–57.
3. On the greatest order of an element in the symmetric group, Amer. Math
Monthly 79 (1972), 500–501.
4. Complementing sets of n-tuples of integers, Proc. Amer. Math. Soc. 34
(1972), 71–72.
5. Shift dynamical systems over finite fields, Proc. Amer. Math. Soc. 34
(1972), 591–594.
6. Sums of finite sets of integers, Amer. Math. Monthly 79 (1972),1010–1012
7. Integrals of binary sequences, SIAM J. Appl. Math. 23 (1972), 84–86.
 3

1973
8. On the fundamental domain of a discrete group, Proc. Amer. Math. Soc.
41 (1973), 629–630.
1974
9. Catalan’s equation in K(t), Amer. Math. Monthly 81 (1974), 371–373.
10. Minimal bases and maximal nonbases in additive number theory, J. Number
Theory 6 (1974), 324–333.
11. Approximation by continued fractions, Proc. Amer. Math. Soc. 45 (1974),
323–324.
1975
12. Maximal asymptotic nonbases (with P. Erdős), Proc. Amer. Math. Soc.
48 (1975), 57–60.
13. Products of sums of powers, Math. Mag. 48 (1975), 112–113.
14. Linear recurrences and uniform distribution, Proc. Amer. Math. Soc. 48
(1975), 289–291.
15. An algorithm for partitions, Proc. Amer. Math. Soc. 52 (1975), 121–124
16. Oscillations of bases for the natural numbers (with P. Erdős), Proc. Amer.
Math. Soc. 53 (1975), 253–258
17. Round metric spaces, Amer. Math. Monthly 82 (1975), 738–741.
18. Essential components in discrete groups, Amer. Math. Monthly 82 (1975),
834
1976
19. Polynomial Pell’s equations, Proc. Amer. Math. Soc. 56 (1976), 89–92.
20. Partial products in finite groups, Discrete Math. 15 (1976), 201–203.
21. Partitions of the natural numbers into infinitely oscillating bases and non-
bases (with P. Erdős), Comment. Math. Helv. 51 (1976), 171–182.
22. Piecewise linear functions with almost all points eventually periodic, Proc.
Amer. Math. Soc. 60 (1976), 75–81.
23. Difference operators and periodic sequences over finite modules, Acta Math.
Acad. Sci. Hungar. 28 (1976), 219–224.
24. Mellin’s formula and some combinatorial identities (with S. Chowla), Monatsh.
Math. 81 (1976), 261–265.
25. Prime polynomial sequences (with S. D. Cohen and p. Erdős), J. London
math. Soc. (2) 14 (1976), 559–562.
1977
26. Permutations, periodicity, and chaos, J. Combinatorial Theory Ser. A 22
(1977), 61–68.
27. s-maximal nonbases of density zero, J. London Math. Soc. (2) 15 (1977),
29–34.
28. Nonbases of density zero not contained in maximal nonbases (with P. Erdős),
J. London Math. Soc. (2) 15 (1977), 403–405.
29. Asymptotic distribution and asymptotic independence of sequences of in-
tegers, Acta Math. Acad Sci. Hungar. 29 (1977), 207–218.
30. Oscillations of bases in number theory and combinatorics, in: Number the-
ory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes
in Math., Vol. 626, Springer, Berlin, 1977, pages 217–231.
4

1978
31. Multiplication rules for polynomials, Proc. Amer. Math. Soc. 69 (1978),
210–212.
32. Sets of natural numbers with no minimal asymptotic bases (with P. Erdős),
Proc. Amer. Math. Soc. 70 (1978), 100–102.
33. Monomial congruences, Monatsh. Math. 85 (1978), 199–200.
34. Representation functions of sequences in additive number theory, Proc.
Amer. Math. Soc. 72 (1978), 16–20.
1979
35. Bases and nonbases of squarefree integers (with P. Erdős), J. Number The-
ory. 11 (1979), 197–208.
36. Additive h-bases for lattice points, in: Second International Conference on
Combinatorial Mathematics (New York, 1978), Ann. New York Acad. Sci.
319 (1979), 413–414.
37. Systems of distinct representatives and minimal bases in additive number
theory (with P. Erdős), in: Number theory, Carbondale 1979 (Proc. South-
ern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture
Notes in Math., Vol. 751, Springer, Berlin, 1979, pages 89–107.
38. Classification problems in K-categories, Fund. Math. 105 (1979/80), 187–
197.
1980
39. Sumsets of measurable sets, Proc. Amer. Math. Soc. 78 (1980), 59–63.
40. Connected components of arithmetic graphs, Monatsh. Math. 89 (1980),
219–222.
41. Minimal asymptotic bases for the natural numbers (with P. Erdős), J. Num-
ber Theory 12 (1980), 154–159.
42. Sumsets contained in infinite sets of integers, J. Combin. Theory Ser. A
28 (1980), 150–155.
43. Lagrange’s theorem with N 1/3 squares (with S. L. g. Choi and P. Erdős),
Proc. Amer. Math. Soc. 79 (1980), 203–205.
44. Arithmetic progressions contained in sequences with bounded gaps, Canad.
Math. Bull. 23 (1980), 491–493.
1981
45. Waring’s problem for sets of density zero, in Analytic number theory (Philadel-
phia, Pa., 1980), Lecture Notes in Math., Vol. 899, Springer, Berlin, 1981,
pages 301–310.
46. Lagrange’s theorem and thin subsequences of squares (with (P. Erdős), in:
Contributions to Probability, Academic Press, New York, 1981, pages 3–9.
1983
47. Largest and smallest maximal sets of pairwise disjoint partitions, J. Number
Theory 17 (1983), 103–112.
1984
48. The exact order of subsets of additive bases, in: Number Theory (New
York, 1982), Lecture Notes in Math., Vol. 1052, Springer, Berlin, 1984,
pages 273–277.
 5

1985
49. Cofinite subsets of asymptotic bases for the positive integers (with J. C. M.
Nash), J. Number Theory 20 (1985), 363–372.
1986
50. Divisibility properties of additive bases, Proc. Amer. Math. Soc. 96
(1986), 11–14
51. Waring’s problem for finite intervals, Proc. Amer. Math. Soc. 96 (1986),
15–17.
52. Independence of solution sets in additive number theory (with P. Erdős), in:
Probability, statistical mechanics, and number theory, Adv. Math. Suppl.
Stud., Vol. 9, Academic Press, Orlando, FL, 1986, pages 97–105.
1987
53. A short proof of Cauchy’s polygonal number theorem, Proc. Amer. Math.
Soc. 99 (1987), 22–24
54. An extremal problem for least common multiples, Discrete Math. 64 (1987),
221–228.
55. Multiplicative representations of integers, Israel J. Math. 57 (1987), 129–
136.
56. Thin bases in additive number theory, in: Journées Arithmétiques de Be-
sançon (Besançon, 1985), Astérisque 147-148 (1987), 315–317, 345.
57. Problems and results on minimal bases in additive number theory (with
P. Erdős), in: Number Theory (New York, 1984–1985), Lecture Notes in
Math., Vol. 1240, Springer, Berlin, 1987, pages 87–96.
58. A generalization of the Goldbach-Shnirel’man theorem, Amer. Math. Monthly
94 (1987), 768–771.
59. Sums of polygonal numbers, in: Analytic number theory and Diophan-
tine problems (Stillwater, OK, 1984), Progr. Math., Vol. 70, Birkhäuser
Boston, Boston, 1987, pages 305–316.
1988
60. Sumsets containing infinite arithmetic progressions (with P. Erdős and A.
Sárközy), J. Number Theory 28 (1988), 159–166.
61. Partitions of bases into disjoint unions of bases (with P. Erdős), J. Number
Theory 29 (1988), 1–9.
62. Minimal asymptotic bases with prescribed densities (with P. Erdős), Illinois
J. Math. 32 (1988), 562–574.
63. Simultaneous systems of representatives for families of finite sets, Proc.
Amer. Math. Soc. 103 (1988), 1322–1326.
64. Minimal bases and powers of 2, Acta Arith. 49 (1988), 525–532.
1989
65. On the maximum density of minimal asymptotic bases (with A. Sárközy),
Proc. Amer. Math. Soc. 105 (1989), 31–33.
66. A simple construction of minimal asymptotic bases (with X.-D. Jia), Acta
Arith. 52 (1989), 95–101.
67. Sumsets containing k-free integers, in Number Theory (Ulm, 1987), Lecture
Notes in Math., Vol. 1380, Springer, New York, 1989, pages 179–184.
6

68. Combinatorial pairs, and sumsets contained in sequences, in: Combinato-

rial Mathematics: Proceedings of the Third International Conference (New
York, 1985), Ann. New York Acad. Sci. 555 (1989), 316–319.
69. Additive problems in combinatorial number theory, in: Number Theory
(New York, 1985/1988), Lecture Notes in Math., Vol. 1383, Springer,
Berlin, 1989, pages 123–139.
70. Sumsets containing long arithmetic progressions and powers of 2 (with A.
Sárközy), Acta Arith. 54 (1989), 147–154.
71. Long arithmetic progressions and powers of 2, in Théorie des nombres (Que-
bec, PQ, 1987), de Gruyter, Berlin, 1989, pages 735–739.
72. Additive bases with many representations (with P. Erdős), Acta Arith. 52
(1989), 399–406.
73. Two applications of combinatorics to number theory, in: Graph theory and
its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci.
576 (1989), 408–410.
1990
74. Simultaneous systems of representatives and combinatorial number theory,
Discrete Math. 79 (1990), 197–205.
75. Extremal properties for bases in additive number theory, in: Number The-
ory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, Vol. 51,
North-Holland, Amsterdam, 1990, pages 437–446.
76. Best possible results on the density of sumsets, in: Analytic number theory
(Allerton Park, IL, 1989), Progr. Math., Vol. 85, Birkhäuser Boston,
Boston, 1990, pages 395–403.
1992
77. On a problem of Rohrbach for finite groups, J. Number Theory 41 (1992),
69–76
1993
78. The simplest inverse problems in additive number theory, in: Number theory
with an emphasis on the Markoff spectrum (Provo, UT, 1991), Lecture
Notes in Pure and Appl. Math., Vol. 147, Dekker, New York, 1993, pages
191–206
1994
79. An inverse theorem for sums of sets of lattice points, J. Number Theory 46
(1994), 29–59
80. Addition theorems for σ-finite groups (with X.-D. Jia), in: The Rademacher
legacy to mathematics (University Park, PA, 1992), Contemp. Math., Vol.
166, Amer. Math. Soc., Providence, RI, 1994, pages 275–284.
1995
81. Inverse theorems for subset sums, Trans. Amer. Math. Soc. 347 (1995),
1409–1418.
82. Independence of solution sets and minimal asymptotic bases (with P. Erdős
and P. Tetali), Acta Arith. 69 (1995), 243–258.
83. Adding distinct congruence classes modulo a prime (with N. Alon and I. Z.
Ruzsa), Amer. Math. Monthly 102 (1995), 250–255.
 7

1996
84. The polynomial method and restricted sums of congruence classes (with N.
Alon and I. Z. Ruzsa), J. Number Theory 56 (1996), 404–417.
85. On the sum of the reciprocals of the differences between consecutive primes
(with P. Erdős), in: Number theory (New York, 1991–1995), Springer, New
York, 1996, pages 97–101.
86. Finite graphs and the number of sums and products (with X.-D. Jia), in:
Number theory (New York, 1991–1995), Springer, New York, 1996, pages
211–219.
1997
87. On sums and products of integers, Proc. Amer. Math. Soc. 125 (1997),
9–16.
88. Ballot numbers, alternating products, and the Erdős-Heilbronn conjecture,
in: The mathematics of Paul Erdős, I, Springer, Berliln, 1997, pages 199–
217.
1998
89. Linear forms in finite sets of integers (with S.-P. Han and and C. Kirfel),
Ramanujan J. 2 (1998), 271–281.
1999
90. Inverse theorems and the number of sums and products (with G. Tenen-
baum), in: Structure theory of set addition, Astérisque 258 (1999), 195–204.
91. Number theory and semigroups of intermediate growth, Amer. Math.
Monthly 106 (1999), 666–669.
2000
92. Partitions with parts in a finite set, Proc. Amer. Math. Soc. 128 (2000),
1269–1273.
93. N -graphs, modular Sidon and sum-free sets, and partition identities, Ra-
manujan J. 4 (2000), 59–67.
94. Convexity and sumsets (with G. Elekes and I. Z. Ruzsa), J. Number Theory
83 (2000), 194–201.
95. Growth of sumsets in abelian semigroups, Semigroup Forum 61 (2000),149–
153.
2002
96. Polynomial growth of sumsets in abelian semigroups (with I. Z. Ruzsa), J.
Theor. Nombres Bordeaux 14 (2002), 553–560.
2003
97. Unique representation bases for the integers, Acta Arith. 108 (2003), 1–8.
98. A functional equation arising from multiplication of quantum integers, J.
Number Theory 103 (2003), 214–233.
2004
99. The inverse problem for representation functions of additive bases, in: Num-
ber theory (New York, 2003), Springer, New York, 2004, pages 253–262.
100. On the ubiquity of Sidon sets, in: Number theory (New York, 2003), Springer,
New York, 2004, pages 263–272.
101. Generalized additive bases, Konig’s lemma, and the Erdos-Turan conjec-
ture, J. Number Theory 106 (2004), 70–78.
8

102. Formal power series arising from multiplication of quantum integers, in:
Unusual applications of number theory, DIMACS Ser. Discrete Math. The-
oret. Comput. Sci., Vol. 64, Amer. Math. Soc., Providence, RI, 2004,
pages 145–167.
103. Representation functions of additive bases for abelian semigroups, Int. J.
Math. Math. Sci. (2004), 29-32.
104. Quantum integers and cyclotomy (with A. Borisov and Y. Wang), J. Num-
ber Theory 109 (2004), 120–135.
2005
105. Every function is the representation function of an additive basis for the
integers, Port. Math. (N.S.) 62 (2005), 55–72.
2006
106. Quadratic addition rules for quantum integers (with A. V. Kontorovich),
J. Number Theory 117 (2006), 1–13.
107. A new upper bound for finite additive bases (with S. Gunturk), Acta Arith.
124 (2006),235–255.
108. Additive number theory and the ring of quantum integers, in: General The-
ory of Information Transfer and Combinatorics, Lecture Notes in Computer
Science, Vol. 4123, Springer, Berlin, 2006, pages 505–511.
2007
109 Affine invariants, relatively prime sets, and a phi function for subsets of
{1, 2, . . . , n}, Integers 7 (2007), A1: 1–7.
110. Sets with more sums than differences, Integers 7 (2007), A5: 1–24.
111. Density of sets of natural numbers and the Lévy group (with R. Parikh),
J. Number Theory 124 (2007), 151–158.
112. Linear quantum addition rules, in: Combinatorial Number Theory, de Gruyter,
Berlin, 2007, pages 371–380.
113. Problems in additive number theory, I , in: Additive Combinatorics, Amer.
Math. Soc., Providence, 2007, 263–270.
114. Binary linear forms over finite sets of integers (with K. O’Bryant, B. Orosz,
I. Z. Ruzsa, and M. Silva), Acta Arith. 129 (2007), 341–361.
115. Representation functions of bases for binary linear forms, Funct. Approx.
Comment. Math. 37 (2007), 341–350.
116. Asymptotic estimates for relatively prime subsets of {m + 1, . . . , n} (with
B. Orosz), Integers 7 (2007).
2008
117. Heights in finite projective space, and a problem on directed graphs (with
B. Sullivan), Integers 8 (2008), A13: 1–9.
118. Desperately seeking mathematical truth, Notices Amer. Math Soc. 55:7
(2008), 773.
119. Inverse problems for linear forms over finite sets of integers, J. Ramanujan
Math. Soc., to appear.
120. Inverse problems for representation functions in additive number theory,
Surveys in Number Theory (K. Alladi, ed.), Springer, New York, 2008,
89–117.
121. Tennenbaum at Penn and Rochester, Integers 8:2 (2008), 2–5.
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122. Perfect difference sets constructed from Sidon sets (with J. Cilleruelo),
Combinatorica, to appear.
123. The Caccetta-Haggkvist conjecture and additive number theory, in Analytic
Number Theory: Essays in Honour of Klaus Roth (W. W. L. Chen, W. T.
Gowers, H. Halberstam, W. M. Schmidt, R. C. Vaughan, eds.), Cambridge
University Press, to appear.
124. Maximal Sidon sets and matroids (with J. Dias da Silva), Discrete Math.,
to appear.
125. Heights on the finite projective line, Intern. J. Number Theory, to appear.
126. Sums of products of congruence classes and of arithmetic progressions (with
S. V. Konyagin), Intern. J. Number Theory, to appear.
127. Problems in additive number theory, II: Linear forms and complementing
sets of integers, Journal de Théorie des Nombres de Bordeaux, to appear.
127. Adjoining identities and zeros to semigroups, preprint.
128. Supersequences, rearrangements of sequences, and the spectrum of bases in
additive number theory, J. Number Theory, to appear.
129. Problems in Additive Number Theory, III: Thematic Seminars at the Centre
de Recerca Matemàtica, to appear.
130. Adjoining identities and zeros to semigroups, preprint.
131. Dense sets of integers with prescribed representation functions (with J.
Cilleruelo), preprint.
132. Families of linear semigroups of intermediate growth, preprint.
133. Semidirect products and functional equations for quantum multiplication,
preprint.
134. The spectrum of bases of finite order in additive number theory, preprint.
135. Desperately seeking mathematical proof, preprint.