A Short Character Sum in $\mathbb{F}_{p^3}$

Abstract: We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in $\mathbb{F}{p^3}$ and higher-dimensional lattices in $\mathbb{F}{p^d}$, extending an earlier work of Mei-Chu Chang on Burgess inequality in $\mathbb{F}{p^2}$. In particular, we show that for intervals of size $p^{3/8+\varepsilon}$, the sum $\sum{x, y} \chi(x + \omega y)$, with $\omega \in \mathbb{F}{p^3} \setminus \mathbb{F}_p$, exhibits nontrivial cancellation uniformly in $\omega$. This is further generalized to codimension-one sublattices in $\mathbb{F}{p^d}$, and applied to obtain an alternative estimate for character sums on binary cubic forms.