On the intersection of two homogeneous Beatty sequences
https://doi.org/10.22405/2226-8383-2022-23-5-145-151
Abstract
Homogeneous Beatty sequences are sequences of the form 𝑎𝑛 = [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 𝑆𝛼,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 𝑆𝛼,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:
𝑆𝛼,𝛽(𝑁) = 𝑁/𝛼𝛽 + 𝑂(︀𝑁^((1/2)+𝜀))︀, 𝑁 → ∞.
About the Authors
Alexander Vladimirovich Begunts
Lomonosov Moscow State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
Dmitry Victorovich Goryashin
Lomonosov Moscow State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate
professor
References
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Review
For citations:
Begunts A.V., Goryashin D.V. On the intersection of two homogeneous Beatty sequences. Chebyshevskii Sbornik. 2022;23(5):145-151. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-5-145-151
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