Sets of Integers With No Long Arithmetic Progressions Generated by the Greedy Algorithm
TLDR
In this article, a heuristic formula for the asymptotic density of positive integers containing no arithmetic progression of k terms, generated by the greedy algorithm, was derived for the case where k is composite, and it was shown that for all e > 0, the number of elements of Sk which are less than n is greater than (1 − e)V2n for sufficiently large n.Abstract:
Let Sk be the set of positive integers containing no arithmetic progression of k terms, generated by the greedy algorithm. A heuristic formula, supported by compu- tational evidence, is derived for the asymptotic density of iS^ in the case where k is composite. This formula, with a couple of additional assumptions, is shown to imply that the greedy algorithm would not maximize S^p 1/n over all S with no arithmetic progression of k terms. Finally it is proved, without relying on any conjecture, that for all e > 0, the number of elements of Sk which are less than n is greater than (1 — e)V2n for sufficiently large n.read more
Citations
More filters
Book
Ramsey Theory on the Integers
Bruce Landman,Aaron Robertson +1 more
TL;DR: Van der Waerden's theorem Supersets of $AP$ Subsets of$AP$ Other generalizations of $w(k r)$ Arithmetic progressions (mod $m$) Other variations on van derWaerde's theorem Schur's theorem Rado's theorem Other topics Notation Biobliography Index as discussed by the authors
Journal ArticleDOI
On sequences without geometric progressions
TL;DR: This paper improves on Rankin's results, derive upper bounds, and looks at sequences generated by a greedy algorithm.
Journal ArticleDOI
Greedy algorithm, arithmetic progressions, subset sums and divisibility
TL;DR: A number of density problems for integer sequences with certain divisibility properties and sequences free of arithmetic progressions are considered and sequences generated by a computer using modifications of the greedy algorithm are provided.
Journal ArticleDOI
Greedily partitioning the natural numbers into sets free of arithmetic progressions
TL;DR: The authors decrit un algorithme "glouton" de partition de nombres naturels en ensembles depourvus de progressions arithmetiques de longueur 3.
References
More filters
Journal ArticleDOI
The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms
TL;DR: In this article, it was shown that for each integer k > 3, there exists a set Sk of positive integers containing no arithmetic progression of k terms, such that neSk l/n > (1 e)k log k, with a finite number of exceptional k for each real e > 0.
Journal ArticleDOI
On certain sequences of integers
TL;DR: In this paper, it was shown that there exists an integer s such that the sequence of integers of the form xk1 + + xs has positive density under certain conditions.