Let $D=2^n$ and equip $\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $\pi/\sqrt3$ in probability and in $L^p$ for every $1\le p<\infty$. Quantitatively, it lies within $O(D^{-1/8}(\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\exp\{-\Omega(D^{7/4}\log D)\}$. For each fixed $0\pi/\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\uparrow\pi/\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(\pi^2/3-x^2)^2/(16\zeta(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.
Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<\pi/\sqrt3$, \[ \mu_D\bigl(B_{Q_D}([I],x\sqrt{Q...
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 $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_\delta n^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ Consequ...
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
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
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}....
For $0<d\leq1/4$, let $\mathcal{C}(d)$ be the four-corner Cantor set with Hausdorff dimension $s_d=\log 4/\log(1/d)$. Peres, Simon, and Solomyak, as well as Mattila, asked for which $d$ the measure $\Hau^{s_d}(p_\theta(\mathcal{C}(d)))$ is positive for almost every direction $\theta$. It was open for the range $1/9\leq...
Chong-Wei Liang· 0 citations
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