Research Problems in Function Theory — Problem 7.53

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $|E|$ for Lebesgue measure. In dimension one take any compact $E\subset\mathbb R$ and any assignment $x\mapsto\delta_x>0$, with $I_x=(x-\delta_x,x+\delta_x)$. In dimension $n\ge1$ take any bounded open convex set $K\subset\mathbb R^n$ with $K=-K$ and nonempty interior, and set $K_x=x+\delta_xK$ for $x\in E$, where $E$ is compact. A selection is a finite or countable pairwise disjoint subfamily. Let $c(K)$ be the supremum of constants $c\ge0$ for which every such $E$ and assignment admit a selection satisfying $\sum_j|K_{x_j}|\ge c|E|$. For the two model bodies use the open cube $Q_n=(-1,1)^n$ and the open Euclidean ball $S_n=\{x:|x|<1\}$; dilations are full-dimensional bodies, rather than just their boundary spheres.

Statement recorded in the supplied source.

Determine all admissible constants in the one-dimensional interval problem and, more generally, determine $c(K)$. Find the correct asymptotic behaviour of $c(Q_n)$ and $c(S_n)$ as $n\to\infty$. The source records $1/2$ as admissible and $c>2/3$ as impossible in one dimension, and $2^{-n}\le c(K)<1$ in general. [S2]

Short English statement

Find the sharp fraction of the measure of a compact set that can always be matched by disjoint prescribed centered intervals or dilates of a symmetric convex body, including the large-dimension behaviour for cubes and balls.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307053 (AMR-022-7053). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.53, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.