Publications & Preprints

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Let $A$ be a $n\times n$ random matrix with real-valued, independent, mean-zero, variance-one entries satisfying $\mathbb{E}[a_{ij}^4]\le K$ for some $K>0$, and let $M$ be a fixed invertible $n\times n$ matrix. Writing $\tau_M=\|M^{-1}\|_{\mathrm{HS}}^{-1}$, we prove \[ \mathbb{P}\!\bigl(s_{\min}(MA)\le \varepsilon\tau_M\bigr) \ll \varepsilon+e^{-\Omega(n)} \] for all $\varepsilon\ge 0$, where the implied constants depend only on $K$.

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k\] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3\pi^2}{4}\Big)^{1/3}. \]

Grünbaum’s inequality gives sharp bounds between the volume of a convex body and its part cut off by a hyperplane through the centroid of the body. We provide a generalization of this inequality for hyperplanes that do not necessarily contain the centroid. As an application, we obtain a sharp inequality that compares sections of a convex body to the maximal section parallel to it.