On the Odlyzko-Stanley enumeration problem and Waring's problem over finite fields

We obtain an asymptotic formula on the Odlyzko-Stanley enumeration problem. Let $N_m^{\ast }(k,b)$ be the number of $k$-subsets $S\subseteq F_p^{\ast }$ such that ${\sum }_{x\in S}{x}^m=b$. If $m<p^{1-\delta }$, then there is a constant $ϵ=ϵ(\delta )>0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-\epsilon}+mk-m \choose k}. In addition, let ${\gamma }^{\mathrm{\prime }}(m,p)$ denote the distinct Waring’s number $(\mod p)$,thesmallestpositiveinteger$k$ such that every integer is a sum of m-th powers of $k$-distinct elements $(\mathrm{m}\mathrm{o}\mathrm{d}p)$. The above bound implies that there is a constant $ϵ(\delta )>0$ such for any prime $p$ and any $m<p^{1-\delta }$, if ${ϵ}^{-1}<(e-1)p^{\delta -ϵ}$, then $${\gamma }^{\mathrm{\prime }}(m,p)\le {ϵ}^{-1}\mathrm{.}$$