Littlewood's problem for sets with multidimensional structure

We give $L^1$-norm estimates for exponential sums of a finite sets $A$ consisting of integers or lattice points. Under the assumption that $A$ possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee-Pigno-Smith and Konyagin. These theorems improve upon past work of Petridis.

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