Mean value theorems for rational exponential sums
Doowon Koh, Igor E. Shparlinski
TL;DR
This paper develops finite-field analogues of mean-value theorems for Weyl sums by adapting Mordell's classical method to rational exponential sums over ${\mathbb F}_p$. It shifts focus from pointwise Weyl-sum bounds to precise averaged behavior, introducing restricted mean values $M_{d,k}(\mathfrak U; N)$ over structured sets such as cubes and moment curves. The authors prove new bounds for the even moments $M_{d,2s}(N)$ under the regime $s N^k < p \le s N^{k+1}$, showing $M_{d,2s}(N) \le N^{2s - k(k+1)/2 + o(1)} p^{-d+k}$ for $s \ge d(d+1)/2$, which improves Mordell-type bounds for small $N$ relative to $p$. These results illuminate the distribution of $S(\mathbf{a}; N)$ when averaged over structured coefficient sets, bridging pointwise estimates and full mean-values with potential implications for related exponential-sum problems in finite fields.
Abstract
We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.