Subconvexity in the inhomogeneous cubic Vinogradov system

TL;DR

This paper establishes a subconvexity bound for the number of solutions to an inhomogeneous cubic Vinogradov system, demonstrating a local-global principle under certain conditions and advancing analytical techniques beyond traditional bounds.

Contribution

It provides the first subconvexity results for the inhomogeneous cubic Vinogradov system, extending previous work on homogeneous cases and developing new minor arc estimates.

Findings

Asymptotic formula for solution count under local solubility conditions

Subconvex local-global principle established for the system

Advanced minor arc estimates surpassing square-root cancellation

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.