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On the inhomogeneous Vinogradov system (2110.02366v1)
Published 5 Oct 2021 in math.NT and math.CA
Abstract: We show that the system of equations \begin{align*} \sum_{i=1}s (x_ij-y_ij) = 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.