Equidistributed Sequences

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import FormalConjecturesUtil
/-!
# Equidistributed Sequences

Corollary 4.2 of Chapter 1 states that the sequence $(x^n), n = 1, 2, ... ,$ is equidistributed modulo 1 for
almost all x > 1. And a little bit further down:
"one does not know whether sequences such as $(e^n)$, $(π^n)$, or even $((\frac 3 2)^n)$"
are equidistributed modulo 1 or not.

*References:*
  - [Uniform Distribution of Sequences](https://store.doverpublications.com/products/9780486149998)
by *L. Kuipers* and *H. Niederreiter*, 1974
  - [Wikipedia](https://en.wikipedia.org/wiki/Equidistributed_sequence)
-/

namespace Equidistribution

open scoped Topology

/--
A point `x` is an accumulation point of a sequence `s_0, s_1, ...`
if any neighbourhood of `x` contains a point of the sequence distinct
from `x`.
-/
def IsAccumulationPoint (x : ℝ) (s : ℕ → ℝ) : Prop :=
  x ∈ closure (Set.range s \ {x})

/--
If a point `x` is an accumulation point of a sequence `s_0, s_1, ...` then
there is a subsequence of `s` that tends to `x`
-/
@[category textbook, AMS 11 54]
theorem isAccumulationPoint_exists_subsequence_tendsto
    (x : ℝ) (s : ℕ → ℝ) (hx : IsAccumulationPoint x s) :
    ∃ (u : ℕ → ℕ), StrictMono u ∧ Filter.atTop.Tendsto (s ∘ u) (𝓝 x) := by
  sorry

/--
The sequence `(3/2)^n` is equidistributed modulo `1`.
-/
@[category research open, AMS 11]
theorem isEquidistributedModuloOne_three_halves_pow :
    IsEquidistributedModuloOne (fun n => (3 / 2 : ℝ)^n) := by
  sorry

/-- For any transcendental number `x`, the sequence `x * (3 / 2) ^ n` is
equidistributed modulo 1. -/
@[category research open, AMS 11]
theorem isEquidistributedModuloOne_transcendental_three_halves_pow (x : ℝ)
    (hx : Transcendental ℚ x) :
    IsEquidistributedModuloOne (fun n ↦ x * (3 / 2 : ℝ) ^ n) := by
  sorry

/--
The sequence `(3/2)^n` has infinitely many accumulation points modulo `1`.
-/
@[category research solved, AMS 11]
theorem isAccumulationPoint_three_halves_pow_infinite :
    {x | IsAccumulationPoint x (fun n => Int.fract <| (3 / 2 : ℝ)^n)}.Infinite := by
  sorry

/--
Find an accumulation point of the sequence `(3/2)^n` modulo `1`.
-/
@[category research open, AMS 11]
theorem isAccumulationPoint_three_halves_pow :
    IsAccumulationPoint answer(sorry) (fun n => Int.fract <| (3 / 2 : ℝ)^n) := by
  sorry

/--
There is an accumulation point of the sequence `(3/2)^n` modulo `1`.
-/
@[category test, AMS 11]
theorem isAccumulationPoint_three_halves_pow_exists :
    ∃ p, (IsAccumulationPoint p (fun n => Int.fract <| (3 / 2 : ℝ)^n)) := by
  sorry

end Equidistribution

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.