On sequences without geometric progressions
Authors:
Brienne E. Brown and Daniel M. Gordon
Journal:
Math. Comp. 65 (1996), 17491754
MSC (1991):
Primary 11B05; Secondary 11B83
Published electronically:
October 1, 1996
MathSciNet review:
1361804
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Several papers have investigated sequences which have no term arithmetic progressions, finding bounds on their density and looking at sequences generated by greedy algorithms. Rankin in 1960 suggested looking at sequences without term geometric progressions, and constructed such sequences for each with positive density. In this paper we improve on Rankin's results, derive upper bounds, and look at sequences generated by a greedy algorithm.
 1.
P. Erd\H{o}s and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261264.
 2.
Joseph
L. Gerver and L.
Thomas Ramsey, Sets of integers with nonlong
arithmetic progressions generated by the greedy algorithm, Math. Comp. 33 (1979), no. 148, 1353–1359. MR 537982
(80k:10053), http://dx.doi.org/10.1090/S00255718197905379820
 3.
Joseph
Gerver, James
Propp, and Jamie
Simpson, Greedily partitioning the natural
numbers into sets free of arithmetic progressions, Proc. Amer. Math. Soc. 102 (1988), no. 3, 765–772. MR 929018
(89f:11026), http://dx.doi.org/10.1090/S00029939198809290181
 4.
Richard K. Guy, Unsolved problems in number theory, second ed., SpringerVerlag, 1994. CMP 95:02
 5.
A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell Labs internal memo, 1978.
 6.
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)
 7.
K.
F. Roth, On certain sets of integers, J. London Math. Soc.
28 (1953), 104–109. MR 0051853
(14,536g)
 8.
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)
 1.
 P. Erd\H{o}s and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261264.
 2.
 Joeseph L. Gerver and L. Thomas Ramsey, Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm, Math. Comp. 33 (1979), 13531359. MR 80k:10053
 3.
 Joseph Gerver, James Propp, and Jamie Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions, Proc. Amer. Math. Soc. 102 (1988), 765772. MR 89f:11026
 4.
 Richard K. Guy, Unsolved problems in number theory, second ed., SpringerVerlag, 1994. CMP 95:02
 5.
 A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell Labs internal memo, 1978.
 6.
 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
 7.
 K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104109. MR 14:536g
 8.
 E. Szemerédi, On sets of integers containing no elements in arithmetic progression, Acta Arith. 27 (1975), 199245. MR 51:5547
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Additional Information
Brienne E. Brown
Affiliation:
9211 Mintwood Street, Silver Spring, Maryland 20901
Daniel M. Gordon
Affiliation:
Center for Communications Research, 4320 Westerra Court San Diego, California 92121
Email:
gordon@ccrwest.org
DOI:
http://dx.doi.org/10.1090/S002557189600765X
PII:
S 00255718(96)00765X
Published electronically:
October 1, 1996
