Aleksandrov projection problem for convex lattice sets

Let $K$ and $L$ be origin-symmetric convex integer polytopes in $\mathbb{R}^n$. We study a discrete analogue of the Aleksandrov projection problem. If for every $u\in \mathbb{Z}^n$, the sets $(K\cap \mathbb{Z}^n)|u^\perp$ and $(L\cap \mathbb{Z}^n)|u^\perp$ have the same number of points, is then $K=L$? We give a positive answer to this problem in $\mathbb{Z}^2$ under an additional hypothesis that $(2K\cap \mathbb{Z}^2)|u^\perp$ and $(2L\cap \mathbb{Z}^2)|u^\perp$ have the same number of points.

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