WeakTiling.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 3 contain sorry, 0 declared as axioms

Weak tiling problems

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

Reference:

Geometric implications of weak tiling

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

Imports (1)

FormalConjecturesUtil

def WeakTiling.IsUnionOfNIntervals Documented declaration: IsUnionOfNIntervals line 43 source
Source documentation
A set $A \subseteq \mathbb{R}$ is a union of $n$ intervals if it can be written as $A = \bigcup_{i=1}^{n} (a_i, b_i)$ with $a_1 < b_1 < a_2 < b_2 < \dots < a_n < b_n$, i.e., as a disjoint union of $n$ nondegenerate open intervals in strict order with strict separation. This matches the convention of the paper (see e.g. Theorem 3.4 there).

L4DEL explanation
A set $A \subseteq \mathbb{R}$ is a union of $n$ intervals if it can be written as $A = \bigcup_{i=1}^{n} (a_i, b_i)$ with $a_1 < b_1 < a_2 < b_2 < \dots < a_n < b_n$, i.e., as a disjoint union of $n$ nondegenerate open intervals in strict order with strict separation. This matches the convention of the paper (see e.g. Theorem 3.4 there).
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Real
Proof uses: Exists, Fin, And, LT.lt, Eq, Set.iUnion, Set.Ioo
3 instances, coercions and auxiliary constants left out.
```
def IsUnionOfNIntervals (n : ℕ) (A : Set ℝ) : Prop :=
…
```
def WeakTiling.IsFiniteUnionOfIntervals A set $A \subseteq \mathbb{R}$ is a finite union of intervals if it is a union of $n$ intervals for some $n$ line 51 source
Source documentation
A set $A \subseteq \mathbb{R}$ is a finite union of intervals if it is a union of $n$ intervals for some $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Exists, Nat, WeakTiling.IsUnionOfNIntervals
```
def IsFiniteUnionOfIntervals (A : Set ℝ) : Prop :=
…
```
def WeakTiling.IsWeakTilingMeasure A positive, locally finite Borel measure line 60 source
Source documentation
A positive, locally finite Borel measure $\nu$ on $\mathbb{R}$ is a weak tiling measure for a bounded measurable set $\Omega \subset \mathbb{R}$ if the convolution $1_\Omega \ast \nu = 1_{\Omega^c}$ holds almost everywhere, i.e., $\int 1_\Omega(x - t) \, d\nu(t) = 1_{\Omega^c}(x)$ for a.e. $x \in \mathbb{R}$.
This is Definition 1.1 from the paper.

L4DEL explanation
A positive, locally finite Borel measure $\nu$ on $\mathbb{R}$ is a weak tiling measure for a bounded measurable set $\Omega \subset \mathbb{R}$ if the convolution $1_\Omega \ast \nu = 1_{\Omega^c}$ holds almost everywhere, i.e., $\int 1_\Omega(x - t) \, d\nu(t) = 1_{\Omega^c}(x)$ for a.e. $x \in \mathbb{R}$. This is Definition 1.1 from the paper.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real, MeasureTheory.Measure
Proof uses: And, Bornology.IsBounded, MeasurableSet, MeasureTheory.IsLocallyFiniteMeasure, Filter.Eventually, Eq, MeasureTheory.integral, Set.indicator, OfNat.ofNat, HSub.hSub, Compl.compl, MeasureTheory.ae, MeasureTheory.MeasureSpace.volume
21 instances, coercions and auxiliary constants left out.
```
def IsWeakTilingMeasure (Ω : Set ℝ) (ν : Measure ℝ) : Prop :=
…
```
def WeakTiling.IsProperTiling Documented declaration: IsProperTiling line 71 source
Source documentation
A proper tiling of $\Omega^c$ by translates of $\Omega$ is specified by a set of translation parameters $T \subseteq \mathbb{R}$ such that the sum of Dirac masses on $T$ is a weak tiling measure for $\Omega$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: WeakTiling.IsWeakTilingMeasure, MeasureTheory.Measure.sum, MeasureTheory.Measure.dirac, Membership.mem
4 instances, coercions and auxiliary constants left out.
```
def IsProperTiling (Ω T : Set ℝ) : Prop :=
…
```
def WeakTiling.HasBoundedDensity Documented declaration: HasBoundedDensity line 77 source
Source documentation
A set $\Lambda \subseteq \mathbb{R}$ has bounded density if the number of points of $\Lambda$ in any unit open interval is uniformly bounded: $\sup_{x \in \mathbb{R}} \#(\Lambda \cap (x, x + 1)) < \infty$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Exists, Nat, And, Set.Finite, Inter.inter, Set.Ioo, HAdd.hAdd, OfNat.ofNat, LE.le, Set.ncard
7 instances, coercions and auxiliary constants left out.
```
def HasBoundedDensity (Λ : Set ℝ) : Prop :=
…
```
theorem WeakTiling.problem_4_1 sorry line 89 source
depends on sorry what this means
Statement uses: Iff, True, Set, Real, WeakTiling.IsFiniteUnionOfIntervals, MeasureTheory.Measure, WeakTiling.IsWeakTilingMeasure, WeakTiling.HasBoundedDensity, MeasureTheory.Measure.support
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem WeakTiling.problem_4_2 Problem 4.2 sorry line 98 source
Source documentation
Problem 4.2. Let $\Omega \subset \mathbb{R}$ be a finite union of three or more intervals. If $\Omega$ weakly tiles its complement, must it also tile its complement properly?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Set, Real, WeakTiling.IsUnionOfNIntervals, MeasureTheory.Measure, WeakTiling.IsWeakTilingMeasure, Exists, WeakTiling.IsProperTiling
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```
Mentions: WeakTiling.IsUnionOfNIntervals, WeakTiling.IsWeakTilingMeasure, WeakTiling.IsProperTiling
theorem WeakTiling.problem_4_3 sorry line 121 source
depends on sorry what this means
Statement uses: Iff, False, Set, Real, WeakTiling.IsFiniteUnionOfIntervals, MeasureTheory.Measure, WeakTiling.IsWeakTilingMeasure, Exists, Nat, NNReal, And, WeakTiling.IsProperTiling, Eq, tsum, SummationFilter.unconditional, OfNat.ofNat, MeasureTheory.Measure.sum, HSMul.hSMul, ENNReal, MeasureTheory.Measure.dirac and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
39 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.