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We prove sufficient conditions for the $\\ell^{p}$-improving inequality \\begin{equation*} \\|A_N f\\|_{\\ell^q(\\mathbb{Z})} \\lesssim_{P,p,q} N^{-d(\\frac{1}{p}-\\frac{1}{q})} \\|f\\|_{\\ell^p(\\mathbb{Z})}, \\qquad N \\in\\mathbb{N}, \\end{equation*} where $1\\leq p \\leq q \\leq \\infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. 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We prove sufficient conditions for the $\\ell^{p}$-improving inequality \\begin{equation*} \\|A_N f\\|_{\\ell^q(\\mathbb{Z})} \\lesssim_{P,p,q} N^{-d(\\frac{1}{p}-\\frac{1}{q})} \\|f\\|_{\\ell^p(\\mathbb{Z})}, \\qquad N \\in\\mathbb{N}, \\end{equation*} where $1\\leq p \\leq q \\leq \\infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. 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