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- W3100946934 abstract "For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. 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 inmathbb{N}, end{equation*} where $1leq p leq q leq infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth." @default.
- W3100946934 created "2020-11-23" @default.
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- W3100946934 date "2020-05-28" @default.
- W3100946934 modified "2023-10-13" @default.
- W3100946934 title "Improving Estimates for Discrete Polynomial Averages" @default.
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- W3100946934 doi "https://doi.org/10.1007/s00041-020-09748-4" @default.
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