A problem of Rankin on sets without geometric progressions

• Autorzy: Melvyn B. Nathanson , Kevin O'Bryant
• Języki publikacji: 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.

Kategorie tematyczne

• 11B75: Other combinatorial number theory
• 05D10: Ramsey theory
• 11B83: Special sequences and polynomials
• 11B25: Arithmetic progressions
• 11B05: Density, gaps, topology

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 170
• Numer: 4
• Strony: 327-342

Daty

wydano

2015

Twórcy

autor

Melvyn B. Nathanson

• Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468, U.S.A.

autor

Kevin O'Bryant

• Department of Mathematics, College of Staten Island (CUNY), Staten Island, NY 10314, U.S.A.

Identyfikatory

DOI

10.4064/aa170-4-2