A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson ; Kevin O'Bryant

Acta Arithmetica, Tome 168 (2015), p. 327-342 / Harvested from The Polish Digital Mathematics Library

Résumé

A geometric progression of length k and integer ratio is a set of numbers of the form ${a,ar,...,ar^{k-1}}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{∞}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃ _{i=1}^{∞} (a_{2i}, a_{2i-1}]$ 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 $(A_i)_{i=1}^{∞}$ of positive integers with A₁ = 1 such that $a_i = 1/A_i$ for all i = 1,2,.... The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of 1,...,n that contains no geometric progression of length k and integer ratio.

Détail

Publié le : 2015-01-01

Zbl 06480402

EUDML-ID : urn:eudml:doc:279278