On a Hybrid Version of the Vinogradov Mean Value Theorem

Abstract

Given a family $\phi =({\phi }_1,\dots ,{\phi }_d)\in \mathbb{Z}[T{]}^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$M_{k,\rho }(\phi ,N)={\int }_{\mathbf{x}\in [0,1{]}^k}\underset{\mathbf{y}\in [0,1{]}^{d-k}}{\sup }{\left|S_{\phi }(\mathbf{x},\mathbf{y};N)\right|}^{\rho }d\mathbf{x}$$ of exponential sums $$S_{\phi }(\mathbf{x},\mathbf{y};N)=\sum _{n=1}^N\exp \left(2\pi i\left(\sum _{j=1}^k{x}_j{\phi }_j(n)+\sum _{j=1}^{d-k}{y}_j{\phi }_{k+j}(n)\right)\right),$$ where $\mathbf{x}=(x_1,\dots ,x_k)$ and $\mathbf{y}=(y_1,\dots ,y_{d-k})$. The case of polynomials ${\phi }_i(T)=T^i$, $i=1,\dots ,d$ 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 ${\sup }_{\mathbf{y}\in [0,1{]}^{d-k}}\left|S_{\phi }(\mathbf{x},\mathbf{y};N)\right|$ for almost all $\mathbf{x}\in [0,1{]}^k$, we obtain nontrivial bounds on $M_{k,\rho }(\phi ,N)$.

16 pages. arXiv admin note: text overlap with arXiv:1903.07330