The extended main conjecture for large measurable sets in Vinogradov's mean value theorem

Let $k\in\mathbb N$, let ${\mathfrak B}\subseteq[0,1)^k$ be measurable, and let $f_k({\boldsymbol\alpha};X)$ denote the Vinogradov exponential sum. Write $\mathrm{mes}({\mathfrak B})$ for the measure of ${\mathfrak B}$.

The large-set extension. Whenever $s>0$ and

$$\mathrm{mes}({\mathfrak B})\gg X^{1-k(k+1)/4},$$

one has

$$\int_{\mathfrak B}|f_k({\boldsymbol\alpha};X)|^{2s}\,\mathrm d{\boldsymbol\alpha}\ll X^\varepsilon\left(X^s\mathrm{mes}({\mathfrak B})+X^{2s-k(k+1)/2}\right).$$

This is presented as a cheaper version applicable for all positive $s$ when the measurable set is not too small. It avoids the convergence restriction on $s$ in the extended main conjecture.