On the inhomogeneous Vinogradov system

We show that the system of equations \begin{align*} \sum_{i=1}^s (x_i^j-y_i^j) = a_j \qquad (1 \le j \le k) \end{align*} has appreciably fewer solutions in the subcritical range $s < k(k+1)/2$ than its homogeneous counterpart, provided that $a_\ell \neq 0$ for some $\ell \le k-1$. Our methods use Vinogradov's mean value theorem in combination with a shifting argument.