kth powers in a generalization of Piatetski-Shapiro sequences

The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c > 1 $, and $ \beta $ are real numbers. The focus of the stud...

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Bibliographic Details
Published inAIMS mathematics Vol. 8; no. 9; pp. 22411 - 22418
Main Author Yukai Shen
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2023
Subjects
asymptotic formula
exponential sum
kth power
piatetski-shapiro sequence
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Summary:The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c > 1 $, and $ \beta $ are real numbers. The focus of the study is on solving equations of the form $ \left\lfloor \alpha n^c +\beta\right\rfloor = s m^k $, where $ m $ and $ n $ are positive integers, $ 1 \leq n \leq N $, and $ s $ is an integer. Bounds for the solutions are obtained for different values of the exponent $ k $, and an average bound is derived over $ k $-free numbers $ s $ in a given interval.
ISSN:2473-6988
DOI:10.3934/math.20231143