Density of Chebyshev nodes

Question

While reading some notes, I came across the following statement:

``Chebyshev points have density $\mu(x) = \frac{N}{\pi\sqrt{1-x^2}}$".

I would like to understand where this formula comes from. My current understanding is that the formula is saying that for Chebyshev points $z_j = (\cos(\pi j/N),\sin(\pi j/N))$, with $N$ large, the number of points after projecting onto the x-axis in the interval $[a,b]$ is given by $$ \int_a^b\frac{N}{\pi\sqrt{1-x^2}}{\rm d}x. $$ How is this formula derived?

As a follow up, if I am given some family of polynomials $p_N(x)$ with $N$ distinct real roots, is there some way to compute the number of roots in the interval $[a,b]$ when $N$ is large?

I am aware of the following question and answer, and while it did present a calculation of the above formula, it did not explain why the presented method is used.

Answer 1

The Chebyshev nodes of order $N$ are the $N$ roots $\{x_k\}_{1\leq k\leq N}$ of the Chebyshev polynomial of the first kind $T_N$. Remember that $T_N(x_k)=\cos(N x_k)=0$ and for $1\leq k \leq N$, $$x_k=\cos\left( \frac{(2k-1)\pi}{2N}\right)\tag{1}$$ How many such roots fall within an interval $(a, b)\subset (-1, 1)$?

Those would be the roots $x_k$ such that $a\leq x_k\leq b$, which, using $(1)$ are for $k$ verifying $$1+\frac{N}{\pi}\arccos b\leq k \leq 1+\frac{N}{\pi}\arccos a$$ That number is $$n_N(a, b)=\frac N \pi \left(\arccos a - \arccos b\right)+\mathcal O(1)$$ where $\mathcal O(1)$ denotes a function that's bounded independently of $N$, $a$ or $b$ (in fact that function is bounded by $1$).

Now, letting $b\rightarrow a$, we obtain that the density is given by $$\boxed{\lim_{b\rightarrow a} \frac{n_N(a, b)}{b-a} = \frac N \pi \lim_{b\rightarrow a} \frac{\arccos a - \arccos b}{b-a}=\frac{N}{\pi\sqrt{1-a^2}}}$$ where we're only considering the first order term for large values of $N$.

For the case of a general polynomial, the same reasoning can be applied so long as you have some formula giving you the distribution of the roots, and that you can invert that formula. That's pretty rare though (most of the time, we don't have a parametrization of the roots of interesting polynomials).