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 $N_m^*(k,b)$ be the number of $k$-subsets $S\subseteq F_p^*$ such that $\sum_{x\in S}x^m=b$. If $m<p^{1-δ}$, then there is a constant $ε=ε(δ)>0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-ε}+mk-m \choose k}. In addition, let $γ'(m,p)$ denote the distinct Waring's number $(\mod p)$, the smallest positive integer $k$ such that every integer is a sum of m-th powers of $k$-distinct elements $(\mod p)$. The above bound implies that there is a constant $ε(δ)>0$ such for any prime $p$ and any $m<p^{1-δ}$, if $ε^{-1}<(e-1)p^{δ-ε}$, then $γ'(m,p)\leq ε^{-1}.$

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