Subconvexity in the inhomogeneous cubic Vinogradov system

Abstract

When $\mathbf{h}\in {\mathbb{Z}}^3$, denote by $B(X;\mathbf{h})$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ and appropriate local solubility conditions on $\mathbf{h}$ are met, we obtain an asymptotic formula for $B(X;\mathbf{h})$, thereby establishing a subconvex local-global principle in the inhomogeneous cubic Vinogradov system. We obtain similar conclusions also when $h_1=0$, $h_2\ne 0$ and $X$ is sufficiently large in terms of $h_2$. Our arguments involve minor arc estimates going beyond square-root cancellation.

arXiv comment18 pages