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import FormalConjecturesUtilLonely runner conjecture
namespace LonelyRunnerConjecture
Consider $n$ runners on a circular track of unit length. At the initial time $t = 0$, all runners are at the same position and start to run; the runners' speeds are constant, all distinct, and may be negative. A runner is said to be lonely at time $t$ if they are at a distance (measured along the circle) of at least $\frac 1 n$ from every other runner. The lonely runner conjecture states that each runner is lonely at some time, no matter the choice of speeds.
@[category research open, AMS 11]
theorem lonely_runner_conjecture (n : ℕ)
(speed : Fin n ↪ ℝ) (lonely : Fin n → ℝ → Prop)
(lonely_def :
∀ r t, lonely r t ↔
∀ r2 : Fin n, r2 ≠ r →
dist (t * speed r : UnitAddCircle) (t * speed r2) ≥ 1 / n)
(r : Fin n) : ∃ t ≥ 0, lonely r t := n:ℕspeed:Fin n ↪ ℝlonely:Fin n → ℝ → Proplonely_def:∀ (r : Fin n) (t : ℝ), lonely r t ↔ ∀ (r2 : Fin n), r2 ≠ r → dist ↑(t * speed r) ↑(t * speed r2) ≥ 1 / ↑nr:Fin n⊢ ∃ t ≥ 0, lonely r t
All goals completed! 🐙
For an $n$-tuple of distinct integer velocities $v_1,\dots,v_n$,
deltaTuple v is the maximal value of $\min_i |t v_i|_{\mathbb{R}/\mathbb{Z}}$ over time.
noncomputable def deltaTuple {n : ℕ} (v : Fin n → ℤ) : ℝ :=
sSup { δ : ℝ | ∃ t : AddCircle (1 : ℝ), ∀ i : Fin n, δ ≤ dist (v i • t : AddCircle (1 : ℝ)) 0 }
The $n$th deltaTuple
over all $n$-tuples of distinct nonzero integer velocities.
noncomputable def deltaGap (n : ℕ) : ℝ :=
sInf { d : ℝ | ∃ v : Fin n ↪ ℤ, (∀ i : Fin n, v i ≠ 0) ∧ d = deltaTuple v }
Theorem 1.3 (Tao, 2017; arXiv:1701.02048). There exists an absolute constant $c > 0$ such that for all sufficiently large $n$, the gap of loneliness satisfies $\delta_n \ge \frac{1}{2n} + \frac{c \log n}{n^2 (\log \log n)^2}$.
@[category research solved, AMS 11]
theorem lonely_runner_conjecture.variants.tao_2017 :
∃ c : ℝ, 0 < c ∧
(∀ᶠ n : ℕ in Filter.atTop,
deltaGap n ≥
((1 : ℝ) / (2 * (n : ℝ))
+ c * Real.log (n : ℝ) /
((n : ℝ) ^ 2 * (Real.log (Real.log (n : ℝ))) ^ 2))) := ⊢ ∃ c,
0 < c ∧
∀ᶠ (n : ℕ) in Filter.atTop, deltaGap n ≥ 1 / (2 * ↑n) + c * Real.log ↑n / (↑n ^ 2 * Real.log (Real.log ↑n) ^ 2)
All goals completed! 🐙
end LonelyRunnerConjecture