As we know that the relation between sum of squares and squares of sums is
$$f(N,X):=\sum_{i=1}^N x_i^2\leq (\sum_{i=1}^N x_i)^2\leq N\sum_{i=1}^N x_i^2:=g(N,X)$$
where $x_i\geq 0$. Resource for example here.
Thus in other words, $$1\leq \frac{(\sum_{i=1}^N x_i)^2}{\sum_{i=1}^N x_i^2}:=\frac{\|X\|_1^2}{\|X\|_2^2}\leq N$$ We denote the lower bound as $f(N,X)$ and upper bound as $g(N,X)$.
We could improve the upper bound by knowing the number of non-zero $x_i$ to $\sqrt{q}$, according to here.
Let $x_i:=(\theta_p - \theta_q) \sin [ (\theta_p - \theta_q) a]$ where $p,q\in\{1,\cdots,m\}$, $m^2=N$.
My question is, which condition we can put on $(\theta,a)$, or say, how to partition the vector space $\mathbb{R}^N$ into subspaces, such that the lower bound $f(N,X)$ of the ratio whose order is higher than $O(1)$, for example, improve it to be $\sqrt{N}$.
