Burgess-type character sum estimates over generalized arithmetic progressions of rank $2$

We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank $2$ in prime fields $\mathbb{F}_p$. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. A key step in our argument involves establishing new upper bounds for the sizes of Bohr sets, which may be of independent interest.

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