arXiv subjects

Brayden Letwin

Publications and source records attributed to Brayden Letwin.

5 recordsLinked to original sources

arXiv2026

The KLS constant is $O(\log^{1/4} n)$

We confirm the Kannan--Lovász--Simonovits conjecture for quadratic forms: if $X \sim μ$ is an isotropic log-concave random vector in $\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \operatorname{Var}_{X \sim μ}(\langle MX,X\rangle) \leq 2\,\mathbb{E}_{X \sim μ}|\nabla\langle MX,X\rangle|^2. $$ As an application, we apply the above to $M=\mathbb{E}_{X \sim μ}(\langle X,θ\rangle X\otimes X)$ for $θ\in S^{n-1}$ and show that the Kannan--Lovász--Simonovits constant $ψ_n$ satisfies $$ ψ_n\leq C\log^{1/4}n $$ for some absolute constant $C>0$.

math.PR↗

arXiv2026

On the smallest singular value of the product of random and deterministic matrices

Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $κ\in (0, 1)$ such that the entries of $A$ satisfy $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert < 1) \le κ. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $κ$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\varepsilon\ge0$, $$ \mathbb{P}!\left(s_{\min}(MA) \le \frac{\varepsilon}{\lVert M^{-1}\rVert_{\mathrm{HS}}}\right) \le C\varepsilon + e^{-cn}. $$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$

math.PR↗

arXiv2026

Dimension-free Gaussian tail estimates for linear functionals on convex bodies

Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $Θ\subset S^{n-1}$ with size $\left|Θ\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $θ\in Θ$ one has \[ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$.

math.MG↗

arXiv2026

On the maxima of Littlewood polynomials on $[-1,1]$

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, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3π^2}{4}\Big)^{1/3}. \]

math.PR↗

arXiv2024

A generalization of Grünbaum's inequality

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.

math.MG↗