Irregular sets of integers generated by the greedy algorithm
Author:
Joseph L. Gerver
Journal:
Math. Comp. 40 (1983), 667676
MSC:
Primary 10L20; Secondary 10H20
DOI:
https://doi.org/10.1090/S00255718198306894802
MathSciNet review:
689480
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: The greedy algorithm was used to generate sets of positive integers containing no subset of the form $\{ x,x + y,x + 2y\}$, $\{ x,x + y,x + 3y\}$, $\{ x,x + 2y,x + 3y\}$, $\{ x,x + 3y,x + 4y\}$, $\{ x,x + 3y,x + 5y\}$, and $\{ x,x + y,x + 2y,x + 3y\}$, respectively. All of these sets have peaks of density in roughly geometric progression.

P. Erdös & P. Turan, "On certain sequences of integers," J. London Math. Soc., v. 11, 1936, pp. 261264.
 Joseph L. Gerver, The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms, Proc. Amer. Math. Soc. 62 (1977), no. 2, 211–214. MR 439796, DOI https://doi.org/10.1090/S00029939197704397969
 Joseph L. Gerver and L. Thomas Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp. 33 (1979), no. 148, 1353–1359. MR 537982, DOI https://doi.org/10.1090/S00255718197905379820 A. M. Odlyzko & R. P. Stanley, "Some curious sequences constructed with the greedy algorithm," unpublished Bell Laboratories report, January 1978. A. M. Odlyzko, private communication.
 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 (1960/61). MR 142526
Retrieve articles in Mathematics of Computation with MSC: 10L20, 10H20
Retrieve articles in all journals with MSC: 10L20, 10H20
Additional Information
Article copyright:
© Copyright 1983
American Mathematical Society