We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in F℃ and higher-dimensional lattices in F㵧, extending an earlier work of Mei-Chu Chang on Burgess inequality in Fℂ. In particular, we show that for intervals of size p^3/8+, the sum ₗ, ₘ χ (x + ωy), with ω F℃ Fₚ, exhibits nontrivial cancellation uniformly in ω. This is further generalized to codimension-one sublattices in F㵧, and applied to obtain an alternative estimate for character sums on binary cubic forms.
A. K. Chattopadhyay (Mon,) studied this question.