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 $(h_1,\ldots ,h_{ k-1})\ne { \mathbf 0}$, 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.
Categories: math.NT