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The Carbery-Wright is a seminal result about the anti-concentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. However, I cannot find any reference where an explicit estimate on the constant B on the r.h.s. of the inequality is given. Knowing this constant is crucial for the application I have in mind. Are there known estimates on it? I should also say that the polynomial I have in mind is of the form $p(g_1,g_2,\ldots,g_k)=\langle g_1\otimes g_2 \ldots\otimes g_k,A g_1\otimes g_2 \ldots\otimes g_k\rangle$, where $A\in\mathbb{R}^{d^k\times d^k}$, $\textrm{tr}(A)=0$, and the $g_i\in \mathbb{R}^d$ are random vectors with i.i.d. gaussian entries. Maybe this special structure helps to obtain better anti-concentration estimates. Any help would be appreciated!

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Have you looked at the original paper by A.Carbery and J.Wright ? Theorem 8 page 244 is the famous inequality with a sharp constant.

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  • $\begingroup$ what do you mean by sharp constant? I do not see an explicit constant in that Theorem. You mean this article: pdfs.semanticscholar.org/e997/… ,right? It only says "There exists an absolute constant C". Do we know what that constant is? $\endgroup$ – user134977 2 days ago
  • $\begingroup$ The way it is stated suggests that C is independent of the polynomial. The constant C is numerical. $\endgroup$ – user69642 2 days ago
  • $\begingroup$ Oh, now I understand the issue. Sure, the constant C is numerical, but my application requires bounding this constant. Note that the statement of the inequality only becomes nontrivial for $\alpha\leq C^{-2d}$, where $d$ is the degree of the polynomial. In the application I have in mind, the degree of the polynomial grows, so the inequality is nontrivial only for exponentially small $\alpha$, with the exponent depending on $C$. That is why I actually want to know $C$ in the inequality. $\endgroup$ – user134977 yesterday
  • $\begingroup$ Then, I guess you should be able to verify its order of magnitude numerically by testing different polynomials in dimension 1 for the Gaussian since it does not depend on the dimension, neither on the degree, neither on the polynomial neither on the measure... $\endgroup$ – user69642 yesterday
  • $\begingroup$ I am not sure this is correct. The statement only tells us that there is a universal lower bound. But there is no guarantee that polynomials in dimension 1 are "representative". I.e., it could be the case that a much better constant $C$ holds for dimension $1$ compared to, say 59. Of course, this gives a lower bound on the constant, but not a universal upper bound. $\endgroup$ – user134977 13 hours ago

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