Packing reciprocal rectangles in a square

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.

For $k\in\mathbb N$, let \[ R_k=[0,1/k]\times[0,1/(k+1)]\subset\mathbb R^2, \qquad Q=[0,1]^2. \] A packing in $Q$ consists of congruent copies $P_k$ of all the $R_k$ such that $P_k\subseteq Q$ and $\operatorname{int}(P_i)\cap\operatorname{int}(P_j)=\varnothing$ whenever $i\ne j$. Thus contacts between boundaries are allowed. Equivalently, each copy has the form $P_k=v_k+U_kR_k$, with $v_k\in\mathbb R^2$ and $U_k\in SO(2)$ a planar rotation. The question concerns one simultaneous packing of the entire infinite sequence. Since \[ \sum_{k=1}^{\infty}\operatorname{area}(R_k) =\sum_{k=1}^{\infty}\left(\frac1k-\frac1{k+1}\right)=1, \] any such packing leaves an uncovered set of planar area zero; a pointwise tiling is not required. Packings of large finite prefixes do not by themselves specify an infinite packing. [S2]

Statement recorded in the supplied source.

Do there exist vectors $v_k$ and rotations $U_k\in SO(2)$ for every $k\in\mathbb N$ such that $v_k+U_kR_k\subseteq Q$ and the interiors of these copies are pairwise disjoint? [S2]

Short English statement

Can one copy of every $1/k$ by $1/(k+1)$ rectangle, starting with $k=1$, fit in a single unit square without overlapping interiors? The total area is exactly one, so a packing would cover the square except for a set of area zero.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3900015 (AMR-038-0015). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mingliang Zhu and Antal Joós, Packing $1.35\cdot10^{11}$ rectangles into a unit square (2022), arXiv:2211.10356. Section 1, pp. 1--2: definition of packing and complete Meir--Moser rectangle sequence; section 2, finite-prefix results; section 3, distinction from the infinite question. https://arxiv.org/abs/2211.10356.