Anti-concentration of polynomials on Haar measure

Question

Let $X\in\mathbb{R}^n$ follow the Haar measure (i.e. uniformly distributed over the unit sphere), and $P$ be a degree-$d$ polynomial such that $\mathrm{Var}[P(X)]=1$. Are there constants $c(n,d)>0$ and $c'(n,d)$ such that $$\Pr[|P(X)|\leq\varepsilon]\leq c'(n,d)\cdot\varepsilon^{c(n,d)}$$ holds for every $\varepsilon\geq 0$?

When $X$ consists of independent random variables, I'm aware of results that provide the above bound, and the constants even depend only on $d$ (for instance the Carbery-Wright inequality when $X$ follows the Gaussian distribution). On the other hand, when $P$ is homogeneous, we can reduce it to the Gaussian case as $X=G/\|G\|_2$ where $G$ is Gaussian and $\|G\|_2^2$ follows $\chi^2(n)$ distribution, and a tail bound on the later will introduce dependence on $n$ to $c'(n,d)$.

However, in the general case when $P$ is not homogeneous, are there any tools to prove such anti-concentration bounds?

I would also like to note that in the actual problem I'm studying, $X$ follows the Haar measure on unitary group, but I think the unit sphere case would be a good starting point. Thanks in advance.

Answer 1

This is slightly too long for a comment, and so I'm posting it as an answer.

It doesn't seem to me like there is a simple "hack" to translate Carbery-Wright to this setting. The original proof of Carbery-Wright is fairly geometric and so doesn't translate nicely to settings like uniform on the unit sphere. However, what you seek is much weaker than what Carbery-Wright guarantees, in that you don't care about the optimal power on $\varepsilon$. For this kind of a statement, there are more combinatorial approaches to anti-concentration. In particular, one way to prove anti-concentration for non-linear functions is to use a "decoupling" approach. For instance, Shachar Lovett has a nice proof of a weaker version of Carbery-Wright in this line (you can download the pdf of his proof here). It seems to me that something like this approach should be plausible for your case. The main idea is that it converts a non-linear anti-concentration statement into a linear one. Even more simply in your case, all you need is to convert a non-homogeneous anti-concentration statement into a homogeneous one. Claims 3.2 and 3.3 in Lovett's paper do this kind of work, which it seems to me should be fairly analogous in your setting.