Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T07:03:06.997Z Has data issue: false hasContentIssue false
  • English
  • Français

XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression*

Published online by Cambridge University Press:  14 February 2012

R. A. Rankin
R. A. Rankin
Affiliation:
Department of Mathematics, University of Glasgow.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n=3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 65 , Issue 4 , 1961 , pp. 332 - 344
Copyright
Copyright © Royal Society of Edinburgh 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES TO LITERATURE

Behrend, F. A., 1946. “On séts of integers which contain no three terms in arithmetical progression”, Proc. Nat. Acad. Sci. Wash., 32, 331–332.Google Scholar
Erdös, P., and Turàn, P., 1936. “On some sequences of integers”,/. Lond. Math. Soc., 11, 261–264.Google Scholar
Landau, E., 1909. Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig, Vol. 1.Google Scholar
Moser, L., 1953. “On non-averaging sets of integers”, Canad. Math. J., 5, 245–252.CrossRefGoogle Scholar
Rankin, R. A., 1961. “On sequences of integers containing no arithmetical progressions”, Bull. Malay. Math. Soc., 8, 43–52.Google Scholar
Rankin, R. A., 1962. “Representations of a number as the sum of a large number of squares”, Proc. Roy. Soc. Edin., A, 65, 318–331.Google Scholar
Roth, K. F., 1952. “Sur quelques ensembles d'entiers”, C. R. Acad. Sci. Paris, 234, 388–390.Google Scholar
Roth, K. F., 1953. “On certain sets of integers—I”, J. Lond. Math. Soc, 28, 104–109.Google Scholar
Roth, K. F., 1954. “On certain sets of integers”11, J. Lond. Math. Soc, 29, 20–26.CrossRefGoogle Scholar
Salem, R., and Spencer, D. C, 1942. “On sets of integers which contain no three terms in arithmetical progression”, Proc. Nat. Acad. Sci. Wash., 28, 561–563.Google Scholar
Salem, R., and Spencer, D. C, 1950. “On sets which do not contain a given number of terms in arithmetical progression”, Nieuw Arch. Wisk., 23, 133–143.Google Scholar