On sequences without geometric progressions

Authors:
Brienne E. Brown and Daniel M. Gordon

Journal:
Math. Comp. **65** (1996), 1749-1754

MSC (1991):
Primary 11B05; Secondary 11B83

DOI:
https://doi.org/10.1090/S0025-5718-96-00765-X

Published electronically:
October 1, 1996

MathSciNet review:
1361804

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Abstract | References | Similar Articles | 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), 261--264.**2.**Joeseph L. Gerver and L. Thomas Ramsey,*Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm*, Math. Comp.**33**(1979), 1353--1359. 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), 765--772. MR**89f:11026****4.**Richard K. Guy,*Unsolved problems in number theory*, second ed., Springer--Verlag, 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), 332--344. MR**26:95****7.**K. F. Roth,*On certain sets of integers*, J. London Math. Soc.**28**(1953), 104--109. MR**14:536g****8.**E. Szemerédi,*On sets of integers containing no elements in arithmetic progression*, Acta Arith.**27**(1975), 199--245. 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:
https://doi.org/10.1090/S0025-5718-96-00765-X

Published electronically:
October 1, 1996