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

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 $p^{3\mathrm{/}8+\epsilon }$, 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.