Geodesic
On the intersection of two homogeneous Beatty sequences
Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 145-151.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{CHEB_2022_23_5_a11,
     author = {A. V. Begunts and D. V. Goryashin},
     title = {On the intersection of two homogeneous {Beatty} sequences},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {145--151},
     publisher = {mathdoc},
     volume = {23},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a11/}
}
TY  - JOUR
AU  - A. V. Begunts
AU  - D. V. Goryashin
TI  - On the intersection of two homogeneous Beatty sequences
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 145
EP  - 151
VL  - 23
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a11/
LA  - ru
ID  - CHEB_2022_23_5_a11
ER  - 
%0 Journal Article
%A A. V. Begunts
%A D. V. Goryashin
%T On the intersection of two homogeneous Beatty sequences
%J Čebyševskij sbornik
%D 2022
%P 145-151
%V 23
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a11/
%G ru
%F CHEB_2022_23_5_a11
A. V. Begunts; D. V. Goryashin. On the intersection of two homogeneous Beatty sequences. Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 145-151. http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a11/

[1] Beatty S., “Problem 3173”, American Mathematical Monthly, 33:3 (1926), 159

[2] Begunts A. V., Goryashin D. V., “Aktualnye zadachi, svyazannye s posledovatelnostyami Bitti”, Chebyshevskii sbornik, 18:4 (2017), 97–105 | DOI

[3] Technau M., On Beatty sets and some generalisations thereof, Würzburg University Press, Würzburg, 2018 | DOI

[4] Skolem Th., “On certain distributions of integers in pairs with given differences”, Math. Scand., 5 (1957), 57–68

[5] Bang T., “On the sequence $[n\alpha]$, $n=1,2,3\ldots$. Supplementary note to the preceding paper by Th. Skolem”, Math. Scand., 5 (1957), 69–76

[6] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lektsii po matematicheskomu analizu, 4-e izd., pererab. i dop., Drofa, M., 2004, 640 pp.

[7] Shmidt Volfgang M., “O sovmestnykh priblizheniyakh dvukh algebraicheskikh chisel ratsionalnymi”, Matematika, 15:3 (1971), 3–25

[8] Keipers L., Niderreiter G., Ravnomernoe raspredelenie posledovatelnostei, Per. s angl., Nauka, M., 1985, 408 pp.

[9] Beck J., “Probabilistic Diophantine Approximation, I. Kronecker Sequences”, Annals of Mathematics, Second Series, 140:2, Sep. (1994), 449+451–502