2024 On the number of representations of integers as differences between Piatetski-Shapiro numbers
Yuuya Yoshida
Funct. Approx. Comment. Math. Advance Publication 1-26 (2024). DOI: 10.7169/facm/240813-26-8

Abstract

For $\alpha>1$, set $\beta=1/(\alpha-1)$. We show that, for every $1<\alpha<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^\alpha}\rfloor - \lfloor{m^\alpha}\rfloor$ is equal to $\beta\alpha^{-\beta}\zeta(\beta)d^{\beta-1} + o(d^{\beta-1})$ as $d\to\infty$, where $\zeta$ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets $(l,m,n)$ of positive integers such that $l<x$ and $\lfloor{l^\alpha}\rfloor + \lfloor{m^\alpha}\rfloor = \lfloor{n^\alpha}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^\alpha}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^\alpha}\rfloor+\lfloor{n_2^\alpha}\rfloor=\lfloor{n_3^\alpha}\rfloor+\lfloor{n_4^\alpha}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_\alpha(N^{4-\alpha})$ when $1<\alpha\le4/3$.

Citation

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Yuuya Yoshida. "On the number of representations of integers as differences between Piatetski-Shapiro numbers." Funct. Approx. Comment. Math. Advance Publication 1 - 26, 2024. https://doi.org/10.7169/facm/240813-26-8

Information

Published: 2024
First available in Project Euclid: 16 December 2024

Digital Object Identifier: 10.7169/facm/240813-26-8

Subjects:
Primary: 11D04 , 11D72 , 11D85
Secondary: 11B25 , 11B30 , 37A44

Keywords: additive energy , arithmetic progression , Discrepancy , equidistribution , Piatetski-Shapiro sequence

Rights: Copyright © 2024 Adam Mickiewicz University

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