Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The sum of the reciprocals of a set of integers with no arithmetic progression of $ k$ terms


Author: Joseph L. Gerver
Journal: Proc. Amer. Math. Soc. 62 (1977), 211-214
MSC: Primary 10L10
MathSciNet review: 0439796
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for each integer $ k \geqslant 3$, there exists a set $ {S_k}$ of positive integers containing no arithmetic progression of k terms, such that $ {\Sigma _{n \in {S_k}}}1/n > (1 - \varepsilon )k\log k$, with a finite number of exceptional k for each real $ \varepsilon > 0$. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin's sets $ \mathcal{A}(k)$, which have the highest known asymptotic density for sets of positive integers containing no arithmetic progression of k terms.


References [Enhancements On Off] (What's this?)

  • [1] H. Davenport and P. Erdös, On sequences of positive integers, Acta Arith. 2 (1937), 147-151.
  • [2] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199-245. MR 51 #5547. MR 0369312 (51:5547)
  • [3] 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 8, 317. MR 0018694 (8:317d)
  • [4] L. Moser, On non-averaging sets of integers, Canad. J. Math. 5 (1953), 245-252. MR 14, 726; erratum p. 1278. MR 0053140 (14:726d)
  • [5] R. A. Rankin, Sets of integers containing not more than a given number of terms in arithmetical progression, Proc. Roy. Soc. Edinburgh Sect. A 65 (1960/61), 332-344. MR 26 #95. MR 0142526 (26:95)
  • [6] Unpublished lecture, Faculté des Sciences, Paris, December 4, 1975.
  • [7] E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canad. Math. Bull. 11 (1968), 409-414. MR 38 #1066. MR 0232743 (38:1066)
  • [8] M. N. Huxley, The distribution of prime numbers, Oxford Math. Monograph, 1972, p. 119. MR 0444593 (56:2943)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10L10

Retrieve articles in all journals with MSC: 10L10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0439796-9
PII: S 0002-9939(1977)0439796-9
Keywords: Arithmetic progression, sum of reciprocals, asymptotic density, Szemerédi's theorem, Rankin's sets
Article copyright: © Copyright 1977 American Mathematical Society