A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson ¹   ; Kevin O'Bryant ²

¹ Department of Mathematics Lehman College (CUNY) Bronx, NY 10468, U.S.A.
² Department of Mathematics College of Staten Island (CUNY) Staten Island, NY 10314, U.S.A.

Acta Arithmetica, Tome 170 (2015) no. 4, pp. 327-342

Résumé

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set $$G^{(k)} = \bigcup_{i=1}^{\infty} (a_{2i} , a_{2i-1} ]$$ contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and integer ratio. It is also proved that there is a strictly increasing sequence $(A_i)_{i=1}^{\infty}$ of positive integers with $A_1 = 1$ such that $a_i = 1/A_i$ for all $i = 1,2,\ldots.$The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of $\{1,\dots,n\}$ that contains no geometric progression of length $k$ and integer ratio.

Détail

DOI : 10.4064/aa170-4-2

Keywords: geometric progression length integer ratio set numbers form dots k positive real number integer geq each integer geq greedy algorithm construct strictly decreasing sequence infty positive real numbers set bigcup infty i contains geometric progression length integer ratio moreover maximal subset contains geometric progression length integer ratio proved there strictly increasing sequence infty positive integers ldots set gives lower bound maximum cardinality subset dots contains geometric progression length integer ratio