Subconvexity in Inhomogeneous Vinogradov Systems

Abstract

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 $$ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\leqslant j\leqslant k), $$ with $1\leqslant x_i,y_i\leqslant X$. When $s\lt k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, 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.