Let $X\in S^{d-1}$ be a uniformly selected random point on the unit sphere.
For high dimensions, we have that each coordinate $X_i$ tends to be distributed as $\mathcal N(0,1/d)$, with a very weak dependence between the coordinates.
Denote by $X_{(1)}\le\ldots\le X_{(d)}$ the ordered coordinates of $X$ and let $c>1$ be a constant.
It is known that $X_{(d)}=\Theta(\sqrt{\log d/d})$ with high probability. That is, the maximal coordinate is expected to be asymptotically larger than most (which would be in $[-\alpha/\sqrt d, +\alpha/\sqrt d]$ for an appropriate choice of $\alpha$).
Let $Y\in\{0,\ldots,d\}$ denote the random variable for which $X_{(d-Y)}\le c / \sqrt d\le X_{(d-Y+1)}$.
That is, $Y$ is the number of coordinates larger than $c / \sqrt d$.
- How can we upper bound $Y$ as a function of $d$?
For example, can we prove that if we remove the $Y=O(\frac{d}{\log^2 d})$ largest coordinates of $X$, the rest will be smaller than $2/\sqrt d$?
(This is an example, and $O(\frac{d}{\log^2 d})$ is useful for my application, but any $o(d)$ bound on $Y$ would be interesting).
Unfortunately, your ideal upper bounds on $Y$ cannot hold. Here is one easy way to see this. Sample a standard Gaussian $g$ in $\mathbb R^d$. By rotational invariance, $g/\| g\|$ is a uniform vector on the sphere. It also turns out that $\| g\|$ is tightly concentrated around $\sqrt{d}$. There are several ways to see this, for example this follows from Gaussian concentration.
But for a standard Gaussian vector, the number of coordinates larger than a constant is simply a binomial with parameters $d$ and $p=\Omega(1)$. For sure, there are at least, say, $dp/100$ such coordinates with high probability. If you adjust all constants properly, this should imply that $Y$ has $\Omega(d)$ coordinates larger than $c/\sqrt{d}$ with high probability.