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Assume that $\\mathbf{f} = \\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \\cdot \\nabla f_k$ are linearly independent. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^R$, let $\\mu$ be an irrational real number, and let $\\eta$ be a positive real number. We consider the values taken by $\\mathbf{f}(x_1 + \\mu, ..., x_n + \\mu)$ for integers $x_1, ..., x_n$. 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Assume that $\\mathbf{f} = \\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \\cdot \\nabla f_k$ are linearly independent. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^R$, let $\\mu$ be an irrational real number, and let $\\eta$ be a positive real number. We consider the values taken by $\\mathbf{f}(x_1 + \\mu, ..., x_n + \\mu)$ for integers $x_1, ..., x_n$. 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