High-dimensional Lehmer problem on Beatty sequences
Let $ q $ be a positive integer. For each integer $ a $ with $ 1 \leqslant a < q $ and $ (a, q) = 1 $, it is clear that there exists one and only one $ \bar{a} $ with $ 1 \leqslant\bar{a} < q $ such that $ a \bar{a} \equiv 1(q) $. Let $ k $ be any fixed integer with $ k \geq 2, 0 &...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-04-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023684?viewType=HTML |
| _version_ | 1797844084368343040 |
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| author | Xiaoqing Zhao Yuan Yi |
| author_facet | Xiaoqing Zhao Yuan Yi |
| author_sort | Xiaoqing Zhao |
| collection | DOAJ |
| description | Let $ q $ be a positive integer. For each integer $ a $ with $ 1 \leqslant a < q $ and $ (a, q) = 1 $, it is clear that there exists one and only one $ \bar{a} $ with $ 1 \leqslant\bar{a} < q $ such that $ a \bar{a} \equiv 1(q) $. Let $ k $ be any fixed integer with $ k \geq 2, 0 < \delta_{i} \leq 1, i = 1, 2, \cdots, k. $ $ r_{n}\left(\delta_{1}, \delta_{2}, \cdots, \delta_{k}, \alpha, \beta, c; q\right) $ denotes the number of all $ k $-tuples with positive integer coordinates $ \left(x_{1}, x_{2}, \ldots, x_{k}\right) $ such that $ 1 \leq x_{i} \leq \delta_{i}q, \left(x_{i}, q\right) = 1, x_{1} x_{2} \cdots x_{k} \equiv c(q) $, and $ x_{1}, x_{2}, \cdots, x_{k-1} \in B_{\alpha, \beta} $. In this paper, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals and give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums. |
| first_indexed | 2024-04-09T17:16:42Z |
| format | Article |
| id | doaj.art-e2a1a5c0873045be9e2aef8b1dce30ab |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T17:16:42Z |
| publishDate | 2023-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-e2a1a5c0873045be9e2aef8b1dce30ab2023-04-20T01:21:20ZengAIMS PressAIMS Mathematics2473-69882023-04-0186134921350210.3934/math.2023684High-dimensional Lehmer problem on Beatty sequencesXiaoqing Zhao0Yuan Yi 1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, ChinaSchool of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, ChinaLet $ q $ be a positive integer. For each integer $ a $ with $ 1 \leqslant a < q $ and $ (a, q) = 1 $, it is clear that there exists one and only one $ \bar{a} $ with $ 1 \leqslant\bar{a} < q $ such that $ a \bar{a} \equiv 1(q) $. Let $ k $ be any fixed integer with $ k \geq 2, 0 < \delta_{i} \leq 1, i = 1, 2, \cdots, k. $ $ r_{n}\left(\delta_{1}, \delta_{2}, \cdots, \delta_{k}, \alpha, \beta, c; q\right) $ denotes the number of all $ k $-tuples with positive integer coordinates $ \left(x_{1}, x_{2}, \ldots, x_{k}\right) $ such that $ 1 \leq x_{i} \leq \delta_{i}q, \left(x_{i}, q\right) = 1, x_{1} x_{2} \cdots x_{k} \equiv c(q) $, and $ x_{1}, x_{2}, \cdots, x_{k-1} \in B_{\alpha, \beta} $. In this paper, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals and give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums.https://www.aimspress.com/article/doi/10.3934/math.2023684?viewType=HTMLthe lehmer problembeatty sequenceexponential sumasymptotic formula |
| spellingShingle | Xiaoqing Zhao Yuan Yi High-dimensional Lehmer problem on Beatty sequences AIMS Mathematics the lehmer problem beatty sequence exponential sum asymptotic formula |
| title | High-dimensional Lehmer problem on Beatty sequences |
| title_full | High-dimensional Lehmer problem on Beatty sequences |
| title_fullStr | High-dimensional Lehmer problem on Beatty sequences |
| title_full_unstemmed | High-dimensional Lehmer problem on Beatty sequences |
| title_short | High-dimensional Lehmer problem on Beatty sequences |
| title_sort | high dimensional lehmer problem on beatty sequences |
| topic | the lehmer problem beatty sequence exponential sum asymptotic formula |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023684?viewType=HTML |
| work_keys_str_mv | AT xiaoqingzhao highdimensionallehmerproblemonbeattysequences AT yuanyi highdimensionallehmerproblemonbeattysequences |