@article{Zhao2023, 
author = {Xiaoqing Zhao and Yuan Yi},
title = {High-dimensional Lehmer problem on Beatty sequences},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {13492-13502},
keywords = {the Lehmer problem, Beatty sequence, exponential sum, asymptotic formula},
url = {https://www.sciopen.com/article/10.3934/math.2023684},
doi = {10.3934/math.2023684},
abstract = {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              a      ¯       with    1  ⩽            a      ¯        <  q such that    a            a      ¯        ≡  1  (  q  ). Let    k be any fixed integer with    k  ≥  2  ,  0  <      δ          i        ≤  1  ,  i  =  1  ,  2  ,  ⋯  ,  k  .        r          n            (                  δ                  1                    ,              δ                  2                    ,      ⋯      ,              δ                  k                    ,      α      ,      β      ,      c      ;      q        )   denotes the number of all    k-tuples with positive integer coordinates        (                  x                  1                    ,              x                  2                    ,      …      ,              x                  k                      )   such that    1  ≤      x          i        ≤      δ          i        q  ,      (                  x                  i                    ,      q        )    =  1  ,      x          1            x          2        ⋯      x          k        ≡  c  (  q  ), and        x          1        ,      x          2        ,  ⋯  ,      x          k      −      1        ∈      B          α      ,      β      . 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.}
}