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## Acta Arithmetica

2015 | 170 | 4 | 327-342
Tytuł artykułu

### A problem of Rankin on sets without geometric progressions

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EN
Abstrakty
EN
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.
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Tom
Numer
Strony
327-342
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Daty
wydano
2015
Twórcy
autor
• Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468, U.S.A.
autor
• Department of Mathematics, College of Staten Island (CUNY), Staten Island, NY 10314, U.S.A.
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