Five-Term and Higher Congruences Involving Arbitrary Sets and Short Intervals Modulo a Prime

We obtain asymptotic formulas for additive congruences $$\sum _{i=1}^r{m}_i{x}_i^{-s}\equiv \lambda (\mathrm{m}\mathrm{o}\mathrm{d}p),$$ where the $m_i$ range over arbitrary subsets of ${\mathbb{F}}_p^{\ast }$ and the $x_i$ over shifted intervals. For five terms, in the balanced case of common cardinality $N$, the asymptotic holds uniformly in $\lambda$ whenever $$N>p^{14\mathrm{/}29+\epsilon },$$ giving a genuine sub-square-root range. The main input is a centered fourth-moment estimate for the associated double exponential sums. The same method yields sub-square-root thresholds for every fixed $r\ge 5$, including $N>p^{8\mathrm{/}17+\epsilon }$ for six terms, with $${\alpha }_r=\frac{1}{3}+\frac{4}{9\sqrt{r}}+O(r^{-1})$$ as $r\to \mathrm{\infty }$.