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

TL;DR

This paper derives an asymptotic formula for the Odlyzko-Stanley enumeration problem and provides bounds on Waring's number over finite fields, advancing understanding of subset sums and power representations in modular arithmetic.

Contribution

It introduces an asymptotic formula for counting specific subset sums and establishes bounds on Waring's number for finite fields under certain conditions.

Findings

Asymptotic formula for N_m^*(k,b) when m<p^{1-δ}.

Bound on Waring's number γ'(m,p) depending on p and m.

Conditions under which the bounds hold with explicit constants.

Abstract

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 $m<p^{1-\delta}$, then there is a constant $\epsilon=\epsilon(\delta)>0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-\epsilon}+mk-m \choose k}. In addition, let $\gamma'(m,p)$ 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 $\epsilon(\delta)>0$ such for any prime $p$ and any $m<p^{1-\delta}$, if $\epsilon^{-1}<(e-1)p^{\delta-\epsilon}$, then $$\gamma'(m,p)\leq \epsilon^{-1}.$$