Acta Mathematica Hungarica ( IF 0.6 ) Pub Date : 2021-02-06 , DOI: 10.1007/s10474-020-01111-9 C. Chen , I. E. Shparlinski
Given a family \(\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d\) of d distinct nonconstant polynomials, a positive integer \(k\le d\) and a real positive parameter \(\rho\), we consider the mean value
$$M_{k, \rho} (\varphi, N) = \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |^\rho \,d{\rm x} $$of exponential sums
$$S_{\varphi}({\rm x}, {\rm y}; N) = \sum_{n=1}^{N} \exp\biggl(2 \pi i\biggl(\sum_{j=1}^k x_j \varphi_j(n)+ \sum_{j=1}^{d-k}y_j\varphi_{k+j}(n)\biggr)\biggr), $$where \({\rm x} = (x_1, \ldots, x_k)\) and \({\rm y} =(y_1, \ldots, y_{d-k})\). The case of polynomials \(\varphi_i(T) = T^i, i =1, \ldots, d\) and \(k=d\) corresponds to the classical Vinaogradov mean value theorem.
Here motivated by recent works of Wooley [14] and the authors [9] on bounds on \({\rm sup}_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |\) for almost all \({\rm x} \in [0,1]^k\), we obtain nontrivial bounds on \(M_{k, \rho} (\varphi, N)\).
中文翻译：
在维诺格拉多夫均值定理的混合版本上
给定d个不同的非恒定多项式的族\（\ varphi =（\ varphi_1，\ ldots，\ varphi_d）\在\ mathbb {Z} [T] ^ d \）中，一个正整数\（k \ le d \）和一个真实的正参数\（\ rho \），我们考虑平均值
$$ M_ {k，\ rho}（\ varphi，N）= \ int _ {{\ rm x} \ in [0,1] ^ k} \ sup _ {{\ rm y} \ in [0,1] ^ {dk}} | S _ {\ varphi}（{\ rm x}，{\ rm y}; N）| ^ \ rho \，d {\ rm x} $$指数和
$$ S _ {\ varphi}（{\ rm x}，{\ rm y}; N）= \ sum_ {n = 1} ^ {N} \ exp \ biggl（2 \ pi i \ biggl（\ sum_ {j = 1} ^ k x_j \ varphi_j（n）+ \ sum_ {j = 1} ^ {dk} y_j \ varphi_ {k + j}（n）\ biggr）\ biggr），$$其中\（{\ rm x} =（x_1，\ ldots，x_k）\）和\（{\ rm y} = {y_1，\ ldots，y_ {dk}）\）。多项式\（\ varphi_i（T）= T ^ i，i = 1，\ ldots，d \）和\（k = d \）的情况对应于经典Vinaogradov平均值定理。
这里是受Wooley [14]和作者[9]在\（{\ rm sup} _ {{\ rm y} \ in [0,1] ^ {dk}} | S _ {\ varphi \ }（{{rm x}，{\ rm y}; N）| \）几乎所有\（{[rm x} \ in [0,1] ^ k \）中，我们获得\（M_ { k，\ rho}（\ varphi，N）\）。
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