Inverse theorems and the number of sums and products

Asterisque

Volumn 258, Issue , 1999, Pages 195-204

  Nathanson, Melvyn B ^a   Tenenbaum, Gérald ^b  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

Author keywords

Additive number theory; Divisors; Freiman's theorem; Inverse theorems; Product sets; Sums and products of integers; Sumsets

Indexed keywords

EID: 0005899174     PISSN: 03031179     EISSN: None     Source Type: Book Series    
DOI: None     Document Type: Article

References (19)

1. Barban M. B. and Vehov P. P., Summation of multiplicative functions of polynomials (in Russian), Mat. Zametki, 5(6):669-680, 1969.
2. English translation: Math. Notes Acad. Sci. USSR, 5, 1969, 400-407.
3. Erdos P., An asymptotic inequality in the theory of numbers (in Russian), Vestnik Leningrad Univ., Serija Mat. Mekh. i Astr., 15(13), 1960, 41-49.
4. Erdos P., Problems and results on combinatorial number theory III, In M. B. Nathanson, editor, Number Theory Day, New York 1976, volume 626 of Lecture Notes in Mathematics, Berlin, Springer-Verlag, 1977, 43-72.
5. Erdos P. and Szemerédi E., On sums and products of integers, In P. Erdos, L. Alpár, G. Halász, and A. Sárközy, editors, Studies in Pure Mathematics, To the Memory of Paul Turán, Birkhäuser Verlag, Basel, 1983, 213-218.
6. Freiman G. A., On the addition of finite sets. I, Izv. Vysh. Zaved. Matematika, 13(6), 1959, 202-213.
7. Freiman G. A., Inverse problems of additive number theory. VI on the addition of finite sets. III Izv. Vysh. Ucheb. Zaved. Matematika, 28(3), 1962, 151-157.
8. Freiman G. A., Foundations of a Structural Theory of Set Addition, volume 37 of Translations of Mathematical Monographs, American Mathematical Society, Providence, 1973.
9. Hall R. R. and Tenenbaum G., Divisors, Number 90 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1988.
10. Lev V. F. and Smeliansky P. Y., On addition of two different sets of integers, Preprint, 1994.
11. Nathanson M. B., The simplest inverse problems in additive number theory, In A. Pollington and W. Moran, editors, Number Theory with an Emphasis on the Markoff Spectrum, Marcel Dekker, 1993, 191-206.
12. Nathanson M. B., On sums and products of integers, submitted, 1994.
13. Nathanson M. B., Additive Number Theory: 2. Inverse Theorems and the Geometry of Sumsets, Graduate Texts in Mathematics. Springer-Verlag, New York, 1995, to appear.
14. Ruzsa I. Z., Generalized arithmetic progressions and sumsets, to appear.
15. Shiu P., A Brun-Titchmarsh theorem for multiplicative functions, J. reine angew. Math., 313, 1980, 161-170.
16. Steinig J., On G. A. Freiman's theorems concerning the sum of two finite sets of integers, In Conference on the Structure Theory of Set Addition, CIRM, Marseille, 1993, 173-186.
17. Tenenbaum G., Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, In Séminaire de Théorie des Nombres, Paris 1981-1982, volume 38 of Progress in Math., Birkhäuser, Boston, 1983, 303-312.
18. Tenenbaum G., Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Math., 51, 1984, 243-263.
19. Vinogradov A. I. and Linnik Yu. V., Estimate of the sum of the number of divisors in a short segment of an arithmetic progression, Uspekhi Mat. Nauk (N.S.), 12, 1957, 277-280.