On a Hybrid Version of the Vinogradov Mean Value Theorem

Abstract

Given a family $φ= (φ_1, \ldots, φ_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $ρ$, we consider the mean value $$ M_{k, ρ} (φ, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ(\mathbf{x}, \mathbf{y}; N) \right|^ρd\mathbf{x} $$ of exponential sums $$ S_φ( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 πi \left(\sum_{j=1}^k x_j φ_j(n)+ \sum_{j=1}^{d-k}y_jφ_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of polynomials $φ_i(T) = T^i$, $i =1, \ldots, d$ and $k=d$ corresponds to the classical Vinaogradov mean value theorem. Here motivated by recent works of Wooley (2015) and the authors (2019) on bounds on $\sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ( \mathbf{x}, \mathbf{y}; N) \right|$ for almost all $\mathbf{x} \in [0,1]^k$, we obtain nontrivial bounds on $M_{k, ρ} (φ, N)$.