The asymptotic density of sequences

Niven, Ivan

doi:10.1090/S0002-9904-1951-09543-9

The asymptotic density of sequences
HTML articles powered by AMS MathViewer

by Ivan Niven PDF

Bull. Amer. Math. Soc. 57 (1951), 420-434

References

Emil Artin and Peter Scherk, On the sum of two sets of integers, Ann. of Math. (2) 44 (1943), 138–142. MR 7909, DOI 10.2307/1968760
F. A. Behrend, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 331–332. MR 18694, DOI 10.1073/pnas.32.12.331
A. S. Besicovitch, On the density of certain sequences of integers, Math. Ann. 110 (1935), no. 1, 336–341. MR 1512943, DOI 10.1007/BF01448032
R. Creighton Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560–580. MR 18196, DOI 10.2307/2371785
Ellen F. Buck and R. C. Buck, A note on finitely-additive measures, Amer. J. Math. 69 (1947), 413–420. MR 20123, DOI 10.2307/2371862
Luther Cheo, On the density of sets of Gaussian integers, Amer. Math. Monthly 58 (1951), 618–620. MR 43836, DOI 10.2307/2306357
S. Chowla, A generalization of Meyer’s theorem on indefinite quadratic forms in five or more variables, J. Indian Math. Soc. (N.S.) 25 (1961), 41. MR 139564
S. D. Chowla and John Todd, The density of reducible integers, Canad. J. Math. 1 (1949), 297–299. MR 30558, DOI 10.4153/cjm-1949-025-4
J. G. van der Corput, On sets of integers. I, II, Nederl. Akad. Wetensch., Proc. 50 (1947), (1947). MR 20584
J. G. van der Corput, On de Polignac’s conjecture, Simon Stevin 27 (1950), 99–105 (Dutch). MR 35298
F. J. Dyson, A theorem on the densities of sets of integers, J. London Math. Soc. 20 (1945), 8–14. MR 15074, DOI 10.1112/jlms/s1-20.1.8
P. Erdös, On the asymptotic density of the sum of two sequences, Ann. of Math. (2) 43 (1942), 65–68. MR 5517, DOI 10.2307/1968880
P. Erdös, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), 685–692. MR 26088, DOI 10.1090/S0002-9904-1948-09057-7
P. Erdös, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), 75–78. MR 29406
Paul Erdös and Ivan Niven, The $\alpha +\beta$ hypothesis and related problems, Amer. Math. Monthly 53 (1946), 314–317. MR 16077, DOI 10.2307/2305501
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Math. Ann. 107 (1933), no. 1, 188–232 (German). MR 1512797, DOI 10.1007/BF01448890
E. Fogels, Zur arithmetik quadratischer Zahlenkörper, Univ. Riga. Wiss. Abh. Kl. Math. Abt. 1 (1943), 23–47 (German, with Latvian summary). MR 0022231
M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. 55 (1949), 641–665. MR 31504, DOI 10.1090/S0002-9904-1949-09242-X
Benjamin Lepson, Certain best possible results in the theory of Schnirelmann density, Proc. Amer. Math. Soc. 1 (1950), 592–594. MR 38367, DOI 10.1090/S0002-9939-1950-0038367-7
U. V. Linnik, On Erdös’s theorem on the addition of numerical sequences, Rec. Math. [Mat. Sbornik] N.S. 10(52) (1942), 67–78 (English, with Russian summary). MR 0007406
Henry B. Mann, A proof of the fundamental theorem on the density of sums of sets of positive integers, Ann. of Math. (2) 43 (1942), 523–527. MR 6748, DOI 10.2307/1968807
Henry B. Mann, On the number of integers in the sum of two sets of positive integers, Pacific J. Math. 1 (1951), 249–253. MR 44562, DOI 10.2140/pjm.1951.1.249
L. Mirsky, A property of square-free integers, J. Indian Math. Soc. (N.S.) 13 (1949), 1–3. MR 30559
Hans-Heinrich Ostmann, Beweis einer Vermutung über die asymptotische Dichte und Verschärfung einer Abschätzung für die Dichte der Summe zweier Zahlenmengen, Deutsche Math. 6 (1941), 213–247 (German). MR 17745
Hans-Heinrich Ostmann, Über die Dichten additiv komponierter Zahlenmengen, Arch. Math. 1 (1949), 393–401 (German). MR 30557, DOI 10.1007/BF02038450
Hans-Heinrich Ostmann, Verfeinerte Lösung der asymptotischen Dichtenaufgabe, J. Reine Angew. Math. 187 (1950), 183–188 (German). MR 34808, DOI 10.1515/crll.1950.187.183
Hans-Heinrich Ostmann, Über die Anzahl der Elemente von Summenmengen, J. Reine Angew. Math. 187 (1950), 222–230 (German). MR 36258, DOI 10.1515/crll.1950.187.222
N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 109 (1934), no. 1, 668–678 (German). MR 1512916, DOI 10.1007/BF01449161
R. Salem and D. C. Spencer, The influence of gaps on density of integers, Duke Math. J. 9 (1942), 855–872. MR 7766, DOI 10.1215/S0012-7094-42-00960-8
R. Salem and D. C. Spencer, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 561–563. MR 7405, DOI 10.1073/pnas.28.12.561
R. Salem and D. C. Spencer, On sets which do not contain a given number of terms in arithmetical progression, Nieuw Arch. Wiskunde (2) 23 (1950), 133–143. MR 0033294
L. Schnirelmann, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933), no. 1, 649–690 (German). MR 1512821, DOI 10.1007/BF01448914
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Trans. Amer. Math. Soc. 39 (1936), no. 2, 315–330. MR 1501849, DOI 10.1090/S0002-9947-1936-1501849-X
Sigmund Selberg, Note on the distribution of the integers $ax^2+by^2+c^{z^{2}}$, Arch. Math. Naturvid. 50 (1949), no. 2, 65–69. MR 31508
Harold N. Shapiro, Some remarks on a theorem of Erdös concerning asymptotic density, Proc. Amer. Math. Soc. 1 (1950), 590–592. MR 38366, DOI 10.1090/S0002-9939-1950-0038366-5

Additional Information

Journal: Bull. Amer. Math. Soc. 57 (1951), 420-434
DOI: https://doi.org/10.1090/S0002-9904-1951-09543-9
MathSciNet review: 0044561