A geometric progression of length k and integer ratio is a set of numbers
of the form {a,ar,β¦,arkβ1} for some positive real number a and
integer rβ₯2. For each integer kβ₯3, a greedy algorithm is used to
construct a strictly decreasing sequence (aiβ)i=1ββ of positive
real numbers with a1β=1 such that the set G(k)=i=1βββ(a2iβ,a2iβ1β] 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
(Aiβ)i=1ββ of positive integers with A1β=1 such that aiβ=1/Aiβ for all i=1,2,3,β¦.
The set G(k) gives a new lower bound for the maximum cardinality of a
subset of the set of integers {1,2,β¦,n} that contains no geometric
progression of length k and integer ratio.Comment: 15 page