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Squares in Piatetski-Shapiro sequences

Volume 181 / 2017

Kui Liu, Igor E. Shparlinski, Tianping Zhang Acta Arithmetica 181 (2017), 239-252 MSC: 11B83, 11K65, 11L07, 11L40. DOI: 10.4064/aa8644-8-2017 Published online: 20 November 2017

Abstract

We study the distribution of squares in a Piatetski-Shapiro sequence $(\lfloor n^c\rfloor)_{n\in\mathbb N}$ with $c \gt 1$ and $c\not\in\mathbb N$. We also study more general equations $\lfloor n^c\rfloor = sm^2$, $n,m\in \mathbb N$, $1\le n \le N$, for an integer $s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.

Authors

  • Kui LiuSchool of Mathematics and Statistics
    Qingdao University
    No. 308, Ningxia Road, Shinan
    Qingdao, Shandong 266071, P.R. China
    e-mail
  • Igor E. ShparlinskiDepartment of Pure Mathematics
    University of New South Wales
    Sydney, NSW 2052, Australia
    e-mail
  • Tianping ZhangSchool of Mathematics and Information Science
    Shaanxi Normal University
    Xi’an, Shaanxi 710119, P.R. China
    e-mail

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