On a hybrid version of the Vinogradov mean value theorem

Abstract

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 $M_{k, \rho} (\varphi, N)$.