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

Abstract

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

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