Weak tiling problems

Problem 4.1. Let Ω ⊂ ℝ be a finite union of intervals and ν a weak tiling measure for Ω. Must supp(ν) have bounded density?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-paper-weak-tiling,
  title        = {Weak tiling problems},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-weak-tiling}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The question

problem_4_1. Problem 4.1. Let $Ω \subset R$ be a finite union of intervals and $ν$ a weak tiling measure for $Ω$ . Must $supp (ν)$ have bounded density?

problem_4_2. Problem 4.2. Let $Ω \subset R$ be a finite union of three or more intervals. If $Ω$ weakly tiles its complement, must it also tile its complement properly?

Problems 4.1, 4.2, and 4.3 from arxiv/2506.23631.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.WeakTiling (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem problem_4_1 :
    answer(sorry) ↔ ∀ (Ω : Set ℝ) (_ : IsFiniteUnionOfIntervals Ω)
      (ν : Measure ℝ) (_ : IsWeakTilingMeasure Ω ν), HasBoundedDensity ν.support

theorem problem_4_2 :
    answer(sorry) ↔ ∀ (n : ℕ) (_ : 3 ≤ n) (Ω : Set ℝ)
      (_ : IsUnionOfNIntervals n Ω) (ν : Measure ℝ) (_ : IsWeakTilingMeasure Ω ν),
      ∃ T : Set ℝ, IsProperTiling Ω T

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Geometric implications of weak tiling

See also FormalConjectures.Wikipedia.Fuglede for Fuglede's spectral set conjecture, which motivates the study of weak tilings.

Source and licence

Imported from Formal Conjectures (research papers), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.