On the Odlyzko-Stanley enumeration problem and Waring's problem over finite fields (Q2444493)

For given real \(x\) and nonnegative integer \(n\) let \[ F(x,n) = x(x-1) \cdots (x-n+1) \] be the falling factorial. Let \(\delta\) be a positive real number. Let \(e\) be \(\exp(1)\). Let \(p\) be an odd prime number. Let \( \mathbb{F}_p\) be the finite field with \(p\) elements. Let \(b \in \mathbb{F}_p\). Let \(N(m,k,b)\) be the number of subsets \(S\) of length \(k\) of \(\mathbb{F}_p\) not containing 0 such that \(b\) is a sum over \(S\) of \(m\)-th powers. The main result of the paper is: (a) If \(m p^{\delta} < p\) then there is an \(\varepsilon \in (0,\delta)\) for which \[ \left | p k! N(m,k,b) - F(p-1,k) \right | \leq p k! F(p^{1- \varepsilon} +mk -m,k). \] (b) If, for some \(c \in (0,1)\), \[ -\frac{\ln(p)}{\ln(c)} < k < p^{\delta}\left(c -\frac{1}{p^{\varepsilon}}\right), \] then we can write \(b\) as the sum of \(k\), two-by-two distinct, \(m\)-th powers. This holds in particular when we take \(k\) such that \( 1 < k \varepsilon < \varepsilon(e-1)p^{\delta - \varepsilon}\). The author's results are applied to the Waring problem with distinct elements over \( \mathbb{F}_p\). An open question proposed is whether we can replace \(m\) by 1 just in the right-hand side of (a) above.