Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\in\mathcal{P}_m(\mathbb{R}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to $\gamma_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<\beta<1/2$, \[ \gamma_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-\beta})M(P) \right\} \longrightarrow1 \] as $n\to\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.
Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\lVert f\rVert_{r}:=\left(\int_{\mathbb{T}}|f|^{r}\,\mathrm{d} m\right)^{1/r}$, with $\lVert f\rVert_{0}$ and $\lVert f\rVert_{\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)...
Let $D_{m,n}\subset\C^{m\times n}$ be the union of the zero loci of the maximal minors of a generic $m\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\cong\GL_m(\C)\times\Gr(m,n)^\circ$, where $\Gr(m,n)^\circ$ is the open stratum of the Grassmannian on which all Pl\"ucker coordinates are non...
Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$ t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m...
We consider the central Hardy--Littlewood maximal operator in $\mathbb{R}^n$. For every integer $n\geq1$, we construct two bounded, nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ that differ on a set of positive measure and satisfy $M_cf(x)=M_cg(x)$ at every point $x\in\mathbb{R}^n$. Thus the central maximal operator...
We establish sharp $L^p$ maximal estimates for averages over smooth nondegenerate curves in $\mathbb R^n$ for $p>n$. More precisely, we show that for any nondegenerate curve $\gamma$ in $\mathbb R^n$ and the associated maximal operator \begin{equation} M f(x):=\sup_{t>0} \Big|\int_{-1}^1f(x-t\gamma(s))\chi(s)\,\mathrm{...
Given $0\le m<n$ and a positive Radon measure $\mu$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(\mu):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}\mu}|G_R^\mu(x)-1|\Bigr],$$ where $G_R^\mu(x):=(4\pi R^2)^{-m/2}\int_{\mat...
Alessandro Pigati· 0 citations
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