Geodesic
On the values of Beatty sequence in an arithmetic progression
Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 364-367.

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In the paper, we consider $N_d(x)=N(x;\alpha,\beta;d,a)$, $x\in\mathbb{N}$, which is the number of values of Beatty sequence $[\alpha n+\beta]$, $1\leqslant n\leqslant x$, for $\alpha>1$ irrational and with bounded partial quotients, $\beta\in[0;\alpha)$, in an arithmetic progression $(a+kd)$, $k\in\mathbb{N}$. We prove the asymptotic formula $N_d(x) = \frac{x}{d} + O(d\ln^3 x)$ as $x\to\infty,$ where the implied constant is absolute. For growing difference $d$ the result is non-trivial provided $d\ll \sqrt{x}\ln^{-3/2-\varepsilon}x$, $\varepsilon>0$.
Keywords: Beatty sequence, arithmetic progression, asymptotic formula. 
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A. V. Begunts; D. V. Goryashin. On the values of Beatty sequence in an arithmetic progression. Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 364-367. http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a23/

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