On the intersection of infinite geometric and arithmetic progressions

Abstract

We prove that the intersection G ∩ A of an infinite geometric progression G = u, uq, uq2, uq3, ..., where u > 0 and q > 1 are real numbers, and an infinite arithmetic progression A contains at most 3 elements except for two kinds of ratios q. The first exception occurs for q = r1/d, where r > 1 is a rational number and d ∈ ℕ. Then this intersection can be of any cardinality s ∈ ℕ or infinite. The other (possible) exception may occur for q = β1/d, where β > 1 is a real cubic algebraic number with two nonreal conjugates of moduli distinct from β and d ∈ ℕ. In this (cubic) case, we prove that the intersection G ∩ A contains at most 6 elements.