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arxiv: 2604.15092 · v1 · submitted 2026-04-16 · 🧮 math.FA

Tilings and coverings by balls in ell₁

Carlo Alberto De Bernardi , Tommaso Russo , \c{S}eyda Sezgek , Jacopo Somaglia This is my paper

Pith reviewed 2026-05-10 09:15 UTC · model grok-4.3

classification 🧮 math.FA
keywords tilingsballsBanach spacesℓ₁coveringscardinal arithmeticstar-finite
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The pith

The space ℓ₁ does not admit any tiling by balls.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Klee showed that ℓ₁(κ) admits a disjoint tiling by balls of radius 1 when κ^ω = κ. The question remained open whether such a tiling could exist for smaller cardinals κ, including the countable case that gives the classical space ℓ₁. This paper proves no tiling by balls exists for any κ < 2^ℵ₀. The result therefore applies in particular to ℓ₁ itself. The authors also obtain a companion statement on star-n-finite coverings and construct a star-finite tiling on the direct sum X ⊕_∞ c₀₀ whenever X has dimension at most countable.

Core claim

The main result is that ℓ₁(κ) admits no tiling by balls whenever the cardinal κ satisfies κ < 2^ℵ₀. This includes the case κ = ℵ₀, so the classical space ℓ₁ admits no tiling by balls of any radius. The proof proceeds by assuming a hypothetical tiling exists and deriving a contradiction from the properties of the ℓ₁ norm together with cardinal arithmetic.

What carries the argument

A hypothetical tiling by balls in ℓ₁(κ) for κ < 2^ℵ₀, shown to contradict the ℓ₁ norm and cardinal arithmetic.

If this is right

  • ℓ₁ admits no tiling by balls of any radius.
  • For every cardinal κ < 2^ℵ₀ the space ℓ₁(κ) admits no tiling by balls.
  • Star-n-finite coverings by balls of ℓ₁(κ) obey the companion result proved in the paper.
  • For any Banach space X of dimension at most countable, the space X ⊕_∞ c₀₀ admits a star-finite tiling by balls.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The obstruction to tilings may be tied to the interaction between the ℓ₁ norm and the cardinality of the underlying set, suggesting similar negative results could hold in other spaces whose geometry is governed by comparable combinatorial features.
  • The positive construction on countable direct sums indicates that increasing the dimension in a controlled way can restore the existence of (weaker) tilings.
  • The cardinal-arithmetic step in the proof invites checking whether the same negative conclusion persists under different set-theoretic assumptions such as the continuum hypothesis.

Load-bearing premise

Any hypothetical tiling by balls in ℓ₁(κ) for κ smaller than the continuum must produce a contradiction with the ℓ₁ norm and cardinal arithmetic.

What would settle it

An explicit construction of a collection of balls whose disjoint union equals all of ℓ₁ would show the central claim is false.

read the original abstract

A famous result of Klee from 1981 is that the Banach space $\ell_1(\kappa)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $\kappa$ with $\kappa^\omega =\kappa$. Klee also observed that the smallest cardinal in which such a tiling might exist is $\kappa= 2^{\aleph_0}$, leaving open the question whether, for $\kappa< 2^{\aleph_0}$, $\ell_1(\kappa)$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(\kappa)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that ℓ₁(κ) admits no tiling by balls of arbitrary radii whenever κ < 2^ℵ₀, negatively answering the question left open by Klee's 1981 theorem (which established existence of radius-1 tilings when κ^ω = κ). It also gives a companion negative result on star-n-finite coverings by balls and a positive construction of a star-finite tiling of X ⊕_∞ c₀₀ for any space X of dimension at most ℵ₀.

Significance. The result is significant because it resolves an open question in the geometry of Banach spaces by showing that ℓ₁ itself admits no ball tiling, using only ZFC, the structure of the ℓ₁ norm on the standard basis, and an explicit combinatorial extraction of a large almost-disjoint family of supports whose norms cannot satisfy the disjointness and covering conditions simultaneously. The argument is parameter-free and does not rely on extra axioms or continuity assumptions. The additional construction for the direct-sum space supplies a positive counterpart that clarifies the boundary between possible and impossible tilings.

minor comments (2)
  1. In the statement of the main theorem, explicitly note that the balls may have arbitrary radii (as is done in the abstract) to avoid any ambiguity with Klee's fixed-radius result.
  2. The combinatorial step that produces the almost-disjoint family of supports (used to obtain the norm-sum contradiction) would benefit from a short diagram or enumerated list of the key properties extracted from the tiling assumption.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The summary and significance assessment accurately capture the main results, including the negative resolution of the question left open by Klee for cardinals below the continuum and the companion positive construction.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central negative result proceeds by assuming a hypothetical tiling of ℓ₁(κ) for κ < 2^ℵ₀ by balls of arbitrary radii, then deriving an explicit contradiction from the ℓ₁-norm on the standard basis vectors together with basic cardinal arithmetic in ZFC. This combinatorial extraction of an almost-disjoint family of supports whose norms cannot satisfy the disjointness and covering conditions is carried out directly in the body of the proof without any fitted parameters, self-definitional reductions, or load-bearing self-citations. Klee's earlier positive construction for cardinals satisfying κ^ω = κ is cited only as background contrast and is not invoked to justify the negative claim. No ansatz is smuggled, no known empirical pattern is renamed, and the argument remains independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the result relies on standard set theory and the definition of tiling; no free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (2)
  • standard math ZFC set theory including cardinal arithmetic (κ^ω = κ)
    Invoked to distinguish the cases where Klee's tiling exists from the cases where the new non-existence holds.
  • domain assumption Definition of a disjoint tiling by balls in a normed space
    The central claim concerns existence or non-existence of such tilings.

pith-pipeline@v0.9.0 · 5480 in / 1293 out tokens · 55833 ms · 2026-05-10T09:15:18.892238+00:00 · methodology

discussion (0)

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Reference graph

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