For an $n$-dimensional convex body $K$, let $\theta_L(K)$ denote its lattice covering density, and let $\Theta_L^{\mathrm{conv}}(n)$ and $\Theta_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantities were known only to lie between a lower bound of order $n$ and an upper bound of order $n^2$, so even their polynomial order was undetermined. We prove that there are absolute constants $c,C>0$ such that \[ c n\log n \le \Theta_L^{\mathrm{sym}}(n) \le \Theta_L^{\mathrm{conv}}(n) \le Cn\log n\,(\log\log n)^{10/3+o(1)}. \] Thus both worst-case quantities are $n\log n\,(\log n)^{o(1)}$, and the upper and lower bounds differ by a factor at most $(\log\log n)^{10/3+o(1)}$. For the upper bound, a vertical--horizontal amplification based on weighted Boolean cubes combines covering estimates for low-codimensional sections into an exact lattice covering of an arbitrary convex body. For the lower bound, a random-slab construction and Poisson witnesses on flat tori show, with positive probability, that the resulting body admits no lattice covering of density below $c n\log n$.
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...
We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small f...
We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\a...
Carlos Beltrán, J. Marzo, J. Ortega-Cerdà· 0 citations
Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\...
A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha...
In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n...
Xiang-Hong Chen, Tong-Ou Yang· 1 citation
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