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Chebyshevskii Sbornik, 2022, Volume 23, Issue 5, Pages 145–151
DOI: https://doi.org/10.22405/2226-8383-2022-23-5-145-151
(Mi cheb1261)
 

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On the intersection of two homogeneous Beatty sequences

A. V. Begunts, D. V. Goryashin

Lomonosov Moscow State University (Moscow)
References:
Abstract: Homogeneous Beatty sequences are sequences of the form $a_n=[\alpha n]$, where $\alpha$ is a positive irrational number. In 1957 T. Skolem showed that if the numbers $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers, then the sequences $[\alpha n]$ and $[\beta n]$ have infinitely many elements in common. T. Bang strengthened this result: denote $S_{\alpha,\beta}(N)$ the number of natural numbers $k$, $1\leqslant k\leqslant N$, that belong to both Beatty sequences $[\alpha n]$, $[\beta m]$, and the numbers $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers, then $S_{\alpha,\beta}(N)\sim \frac{N}{\alpha\beta}$ for $N\to\infty.$
In this paper, we prove a refinement of this result for the case of algebraic numbers. Let $\alpha,\beta>1$ be irrational algebraic numbers such that $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers. Then for any $\varepsilon>0$ the following asymptotic formula holds:
$$S_{\alpha,\beta}(N)=\frac{N}{\alpha\beta}+O\bigl(N^{\frac12+\varepsilon}\bigr), N\to\infty.$$
Keywords: homogeneous Beatty sequence, exponential sums, asymptotic formula.
Received: 15.06.2022
Accepted: 22.12.2022
Document Type: Article
UDC: 511.35, 517.15
Language: Russian
Citation: A. V. Begunts, D. V. Goryashin, “On the intersection of two homogeneous Beatty sequences”, Chebyshevskii Sb., 23:5 (2022), 145–151
Citation in format AMSBIB
\Bibitem{BegGor22}
\by A.~V.~Begunts, D.~V.~Goryashin
\paper On the intersection of two homogeneous Beatty sequences
\jour Chebyshevskii Sb.
\yr 2022
\vol 23
\issue 5
\pages 145--151
\mathnet{http://mi.mathnet.ru/cheb1261}
\crossref{https://doi.org/10.22405/2226-8383-2022-23-5-145-151}
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