Distribution of the difference of an integer and its m-th power mod n over incomplete intervals

Let λ,δ be any real numbers with 0<λ,δ⩽1, n>max{[1λ],[1δ]} and m⩾2 be integers. We will study the distribution of the difference of an integer and its m-th power modulo n over incomplete intervals [1,[λn]]. Let (a)n denotes the integer b with 1⩽b⩽n such that b≡a(modn) for any integer a. DefineSm,n,λ,δ={a:1⩽a⩽λn,(a,n)=1,|a−(am)n|⩽δn}. Some asymptotic formulas for∑a∈Sm,n,λ,δ|a−(am)n|k will be given for any nonnegative real number k.