High-dimensional Lehmer problem on Beatty sequences

Let $q$ be a positive integer. For each integer $a$ with $1⩽a<q$ and $(a,q)=1$, it is clear that there exists one and only one $\overline{a}$ with $1⩽\overline{a}<q$ such that $a\overline{a}\equiv 1(q)$. Let $k$ be any fixed integer with $k\ge 2,0<{\delta }_i\le 1,i=1,2,\cdots ,k.$ $r_n\left({\delta }_1,{\delta }_2,\cdots ,{\delta }_k,\alpha ,\beta ,c;q\right)$ denotes the number of all $k$-tuples with positive integer coordinates $\left(x_1,x_2,\dots ,x_k\right)$ such that $1\le x_i\le {\delta }_{i}q,\left(x_i,q\right)=1,x_1{x}_2\cdots x_k\equiv c(q)$, 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.