Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every $\alpha_1,\ldots,\alpha_n>0$, \[ \mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right]. \] When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a R\'enyi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.
We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the R\'{e}nyi entropy of or...
Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \ri...
Thang Pham, A. Pinamonti, Dung The Tran et al.· 0 citations
Let $n\ge2$ be even, let $\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \[ A_d(\lambda) := \bigl[(\lambda_r-\lambda_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb{N}. \] In 1928, Colombo proved that $\det A_{n-1}(\lambda)\ne0$---and hence $\det...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\te...
Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schr\"odinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\f...
D. Suragan· 0 citations
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