On sequences of projections of the cubic lattice

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In this paper we study sequences of lattices which are, up to similarity, projections of Zⁿ⁺¹ onto a hyperplane v⊥, with v ∈ Zⁿ⁺¹ and converge to a target lattice Λ which is equivalent to an integer lattice. We show a sufficient condition to construct sequences converging at rate O(1/ |v|²/ⁿ) and exhibit explicit constructions for some important families of lattices.