A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, and Kevin O'Bryant. "A problem of Rankin on sets without geometric progressions." Acta Arithmetica 170.4 (2015): 327-342. <http://eudml.org/doc/279278>.

@article{MelvynB2015,
abstract = {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.},
author = {Melvyn B. Nathanson, Kevin O'Bryant},
journal = {Acta Arithmetica},
keywords = {geometric progression-free sequences; Ramsey theory},
language = {eng},
number = {4},
pages = {327-342},
title = {A problem of Rankin on sets without geometric progressions},
url = {http://eudml.org/doc/279278},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Melvyn B. Nathanson
AU - Kevin O'Bryant
TI - A problem of Rankin on sets without geometric progressions
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 4
SP - 327
EP - 342
AB - 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.
LA - eng
KW - geometric progression-free sequences; Ramsey theory
UR - http://eudml.org/doc/279278
ER -