Let $\delta>0$ be fixed, let $m = \lceil(1+\delta)n\rceil$, and let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m = \Gamma(B_1^m)$ we prove $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_\delta n^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ Consequently, for all sufficiently large $n$, $$ R_1(n)\ge c n^{5/8}(\log n)^{-1/8}. $$ The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.
Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upo...
Andrew Lott, Á. Magyar, N. R. Ponagandla· 0 citations
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
We prove that, for every real Lipschitz function $A$ and every integer $n\ge1$, $\|T_{A,n}\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\,\|A'\|_\infty^n$, where $T_{A,n}$ is the $n$-th Calder\'on commutator and $C$ is an absolute constant. The essential tool is the Coifman--McIntosh--Meyer theorem, applied to a holomorp...
Joan Hernández, Joan Mateu, Laura Prat· 0 citations
Let $q_m=2m/(m+1)$ and put \[ \beta_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satis...
Joseph Slote, Chun-Kai Tseng, Alexander Volberg· 1 citation· ⚡1
The quantity $q(A\circ A^{-1})$, the minimum modulus of the eigenvalues of $A\circ A^{-1}$, arises naturally in connection with positive diagonal symmetrizability. For an invertible $\mathbf{M}$-matrix $A$ of order $n$, the classical bounds $\frac{2}{n}\leq q(A\circ A^{-1})\leq 1$ are known. We discuss the sharpness of...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Matthew J. Colbrook, Siavash Sadeghi· 0 citations
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