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## Acta Arithmetica

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## A problem of Rankin on sets without geometric progressions

### Volume 170 / 2015

Acta Arithmetica 170 (2015), 327-342 MSC: 11B05, 11B25, 11B75, 11B83, 05D10. DOI: 10.4064/aa170-4-2

#### Abstract

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.

#### Authors

• Melvyn B. NathansonDepartment of Mathematics
Lehman College (CUNY)
Bronx, NY 10468, U.S.A.
e-mail
• Kevin O'BryantDepartment of Mathematics
College of Staten Island (CUNY)
Staten Island, NY 10314, U.S.A.
e-mail

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