Dense subsets of [N]^n contain configurations x, x + r^{m1}e1, ..., x + r^{mn}en for any fixed n and increasing exponents m_i, with density threshold (log N)^{-c}.
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A multidimensional Szemer\'{e}di theorem in integers
Dense subsets of [N]^n contain configurations x, x + r^{m1}e1, ..., x + r^{mn}en for any fixed n and increasing exponents m_i, with density threshold (log N)^{-c}.