On the Odlyzko–Stanley enumeration problem and Waringʼs problem over finite fields

We obtain an asymptotic formula for the Odlyzko–Stanley enumeration problem. Let Nm⁎(k,b) be the number of k  -subsets S⊆Fp⁎ such that ∑x∈Sxm=b∑x∈Sxm=b. If m0ϵ=ϵ(δ)>0 such that|Nm⁎(k,b)−p−1(p−1k)|⩽(p1−ϵ+mk−mk). In addition, let γ′(m,p)γ′(m,p) denote the distinct Waringʼs number (mod p)(mod p), the smallest positive integer k such that every integer is a sum of m-th powers of k   distinct elements (mod p)(mod p). The above bound implies that there is a constant ϵ(δ)>0ϵ(δ)>0 such for any prime p   and any m