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Revisiting and extending a recent result of M.Huxley, we estimate the$L^{p}\left( \mathbb{T}^{d}\right) $ and Weak-$L^{p}\left(\mathbb{T}^{d}\right) $ norms of the discrepancy between the volume and thenumber of integer points in translated domains.

$L^{p}$ and Weak-$L^{p}$ estimates for the number of integer points in translated domains