The sum of the reciprocals of a set of integers with no arithmetic progression of terms
Author:
Joseph L. Gerver
Journal:
Proc. Amer. Math. Soc. 62 (1977), 211214
MSC:
Primary 10L10
MathSciNet review:
0439796
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: It is shown that for each integer , there exists a set of positive integers containing no arithmetic progression of k terms, such that , with a finite number of exceptional k for each real . This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin's sets , which have the highest known asymptotic density for sets of positive integers containing no arithmetic progression of k terms.
 [1]
H. Davenport and P. Erdös, On sequences of positive integers, Acta Arith. 2 (1937), 147151.
 [2]
E.
Szemerédi, On sets of integers containing no 𝑘
elements in arithmetic progression, Acta Arith. 27
(1975), 199–245. Collection of articles in memory of Juriĭ
Vladimirovič Linnik. 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 0018694
(8,317d)
 [4]
Leo
Moser, On nonaveraging sets of integers, Canadian J. Math.
5 (1953), 245–252. 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 (1960/1961), 332–344 (1960/61). 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Unpublished lecture, Faculté des Sciences, Paris, December 4, 1975.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 0232743
(38 #1066)
 [8]
M.
N. Huxley, The distribution of prime numbers, Clarendon Press,
Oxford, 1972. Large sieves and zerodensity theorems; Oxford Mathematical
Monographs. MR
0444593 (56 #2943)
 [1]
 H. Davenport and P. Erdös, On sequences of positive integers, Acta Arith. 2 (1937), 147151.
 [2]
 E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199245. 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), 331332. MR 8, 317. MR 0018694 (8:317d)
 [4]
 L. Moser, On nonaveraging sets of integers, Canad. J. Math. 5 (1953), 245252. 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), 332344. 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), 409414. 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/S00029939197704397969
PII:
S 00029939(1977)04397969
Keywords:
Arithmetic progression,
sum of reciprocals,
asymptotic density,
Szemerédi's theorem,
Rankin's sets
Article copyright:
© Copyright 1977 American Mathematical Society
