On the intersection of two homogeneous Beatty sequences

Abstract

Homogeneous Beatty sequences are sequences of the form 𝑎 𝑛 = [𝛼𝑛], where 𝛼 is a positive irrational number. In 1957 T. Skolem showed that if the numbers 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers, then the sequences [𝛼𝑛] and [𝛽𝑛] have infinitely many elements in common. T. Bang strengthened this result: denote 𝑆 𝛼,𝛽 (𝑁 ) the number of natural numbers 𝑘, 1 𝑘 𝑁 , that belong to both Beatty sequences [𝛼𝑛], [𝛽𝑚], and the numbers 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers, then 𝑆 𝛼,𝛽 (𝑁 ) ∼ 𝑁 𝛼𝛽 for 𝑁 → ∞.In this paper, we prove a refinement of this result for the case of algebraic numbers. Let 𝛼, 𝛽 > 1 be irrational algebraic numbers such that 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers. Then for any 𝜀 > 0 the following asymptotic formula holds:𝑆 𝛼,𝛽 (𝑁 ) = 𝑁 𝛼𝛽 + 𝑂 (︀ 𝑁 1 2 +𝜀 )︀ , 𝑁 → ∞.