Falconer.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Falconer's distance set conjecture

If $d \ge 2$ and $E \subseteq \mathbb{R}^d$ is compact with $\dim_H E > \frac{d}{2}$, then the distance set $$\{ |x - y| \mid x, y \in E \}$$ has positive Lebesgue measure. The restriction $d \ge 2$ is needed; see falconer_conjecture.

References

Imports (1)

FormalConjecturesUtil

lemma Falconer.falconer_conjecture_two Falconer's distance set conjecture, $d = 2$ case sorry line 44 source
Source documentation
Falconer's distance set conjecture, $d = 2$ case.

depends on sorry what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, IsCompact, ENNReal, LT.lt, HMul.hMul, dimH, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Set.image2, Dist.dist
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.
```
lemma falconer_conjecture_two (E : Set <| ℝ²) (hc : IsCompact E) (hd : 2 < 2 * dimH E ) :
    0 < volume (image2 dist E E) := sorry
…
```
lemma Falconer.falconer_conjecture Falconer's distance set conjecture in dimension $d \ge 2$ sorry line 55 source
Source documentation
Falconer's distance set conjecture in dimension $d \ge 2$.
The hypothesis $2 \le d$ excludes a degenerate case: the conclusion is false for $d = 1$. The base-$7$ Cantor set $E = \{\sum_{n \ge 1} a_n 7^{-n} : a_n \in \{0, 1, 2\}\}$ is compact with $\dim_H E = \log 3 / \log 7 > 1/2$, but $E - E$ has digits in $\{-2, \dots, 2\}$, so after fixing $n$ digits it is covered by $5^n$ intervals of length $\frac{2}{3} 7^{-n}$ and is therefore null; its distance set is the image of $E - E$ under $|\cdot|$ and is null too.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Set, EuclideanSpace, Real, Fin, IsCompact, ENNReal, LT.lt, HMul.hMul, dimH, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Set.image2, Dist.dist
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
70 instances, coercions and auxiliary constants left out.
```
lemma falconer_conjecture (d : ℕ) (h2d : 2 ≤ d) (E : Set <| ℝ^d) (hc : IsCompact E)
    (hd : d < 2 * dimH E) :
    0 < volume (image2 dist E E) := sorry
…
```