My question comes from the expression (4.48) of the textbook of Linares and Ponce which states that the direct imitation of Strichartz estimate for $(t,x) \in [0,2\pi]^2$ is not true.
I don't know how to show that counterexample:
There is a constant $c>0$ such that for all large enough integers $N$, $\| \sum_{k=1}^N e^{i (k^2 t + kx)}\|^6_{L^6_{t,x}} \geq c N^3 \log N$.
A direct computation shows that this problem is equivalent to show the following problem of counting lattice point has a lower bound $c N^3 \log N$:
$ \#\{(k_1,j_1,k_2,j_2,k_3,j_3) \in \mathbb{N}^6_{\leq N}: k_1-j_1+k_2-j_2+k_3-j_3 = k_1^2 - j_1^2 + k_2^2 - j_2^2 + k_3^2 - j_3^2 = 0 \} \geq cN^3 \log N $.
My attempt
For each fixed $(j_1,j_2,j_3)$ (maybe each $j_i \geq N/2$ is needed),
I conjecture that the number of lattice points on intersection of the sphere $k_1^2+k_2^2+k_3^2 = j_1^2 + j_2^2+ j_3^2$ and the plane $k_1+k_2+k_3 = j_1 + j_2+ j_3$ is greater than some thing of order $\log N$. But I don't know how to show this.
Many thanks for any discussion or idea!
I find out the above $L^6$ inequality is answered in the notes of Tadairo Oh (maybe earlier in a classical paper of Bouragin ) by a method of computing the Weyl sum $\sum_{n =1}^N e^{i (nx + n^2 t)}$ (sometimes called the exponential sum).
So we can prove that inequality without solving the counting lattice problem I post.
Conversely, I learn that some counting lattice problem can be solved by using suitable Weyl sum.