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 < \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.