Geodesic
Perfect squares of the form $[\alpha n]$
Čebyševskij sbornik, Tome 14 (2013) no. 2, pp. 68-73.

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An asymptotic formula is proved for the number of perfect squares in the sequence $[\alpha n]$ for algebraic numbers $\alpha$ and irrational numbers $\alpha$ with restricted partial quotients.
Keywords: perfect squares, Beatty sequence, asymptotic formula, exponential sums, Weyl sums. 
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     title = {Perfect squares of the form $[\alpha n]$},
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D. V. Goryashin. Perfect squares of the form $[\alpha n]$. Čebyševskij sbornik, Tome 14 (2013) no. 2, pp. 68-73. http://geodesic.mathdoc.fr/item/CHEB_2013_14_2_a5/

[1] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lektsii po matematicheskomu analizu, 3-e izd., pererab. i dop., Drofa, M., 2003

[2] Begunts A. V., “Ob odnom analoge problemy delitelei Dirikhle”, Vestnik Mosk. un-ta. Ser. 1, Matematika. Mekhanika, 2004, no. 6, 52–56

[3] Begunts A. V., “O raspredelenii znachenii summ multiplikativnykh funktsii na obobschennykh arifmeticheskikh progressiyakh”, Chebyshevskii sbornik, 6:2(14) (2005), 52–74 | MR

[4] Abercrombie A. G., “Beatty sequences and multiplicative number theory”, Acta Arith., 70 (1995), 195–207 | MR | Zbl

[5] Abercrombie A. G., Banks W. D., Shparlinski I. E., “Arithmetic functions on Beatty sequences”, Acta Arith., 136:1 (2009), 81–89 | DOI | MR | Zbl

[6] Güloğlu A. M., Nevans C. W., “Sums of multiplicative functions over a Beatty sequence”, Bull. Austral. Math. Soc., 78 (2008), 327–334 | DOI | MR | Zbl

[7] Lü G. S., Zhai W. G., “The divisor problem for the Beatty sequences”, Acta Math. Sinica, 47 (2004), 1213–1216 | MR | Zbl

[8] Vaughan R. C., “On the distribution of $\alpha p$ modulo 1”, Mathematika, 24:48 (1977), 135–141 | DOI | MR