1083.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1083

References:

erdosproblems.com/1083
[APST04] Aronov, Boris and Pach, János and Sharir, Micha and Tardos, Gábor, Distinct distances in three and higher dimensions. Combin. Probab. Comput. (2004), 283--293.
[CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas, Leonidas J. and Sharir, Micha and Welzl, Emo, Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. (1990), 99--160.
[Er46b] Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.
[SoVu08] Solymosi, József and Vu, Van H., Near optimal bounds for the {E}rdős distinct distances problem in high dimensions. Combinatorica (2008), 113--125.

Imports (1)

FormalConjecturesUtil

theorem Erdos1083.erdos_1083 Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances sorry line 46 source
Source documentation
Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that $$f_d(n)=n^{\frac{2}{d}-o(1)}?$$

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, Asymptotics.IsLittleO, Filter.atTop, Filter.Eventually, Eq, minimalDistinctDistances, EuclideanSpace, Fin, ENNReal, HPow.hPow, HSub.hSub, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1083 : answer(sorry) ↔
    ∀ d : ℕ, 3 ≤ d → ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
      ∀ᶠ n : ℕ in atTop,
        (minimalDistinctDistances (ℝ^d) n : ℝ) = (n : ℝ) ^ ((2 : ℝ) / (d : ℝ) - o n) := by
…
```
Mentions: minimalDistinctDistances