Let $k\geq 2$. Consider the following norm of exponenetial sum: $$ I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy. $$ Bourgain mentioned on Page 118 of

https://math.mit.edu/classes/18.158/bourgain-restriction.pdf

that $I(N,6,2)\gtrsim N^3\log N$, where he referenced the following article: https://www.researchgate.net/publication/259308546_The_method_of_trigonometric_sums_in_number_theory.

But I did not find an explicit result in the above article that leads directly to the lower bound above.

So my questions are:

What is the idea to prove the above lower bound? The famous Vinogradov's mean value theorem deals with upper bounds of $I(N,p,2)$, but not lower bounds.

What is a reasonably sharp lower bound for $I(N,p,3)$, or particularly, $I(N,6,3)$? Note that this may not be in the direct form of Vinogradov's mean value theorem, as the $n^2$ term is missing here.