Subconvexity in inhomogeneous Vinogradov systems

Abstract

When $k$ and $s$ are natural numbers and $\mathbf{h}\in {\mathbb{Z}}^k$, denote by $J_{s,k}(X;\mathbf{h})$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When $s<k(k+1)/2$ and $(h_1,\dots ,h_{k-1})\ne 0$, Brandes and Hughes have shown that $J_{s,k}(X;\mathbf{h})=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\mathbf{h})$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.

27 pages