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import FormalConjecturesUtilEquidistributed 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 by L. Kuipers and H. Niederreiter, 1974
namespace Equidistributionopen 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) := x:ℝs:ℕ → ℝhx:IsAccumulationPoint x s⊢ ∃ u, StrictMono u ∧ Filter.Tendsto (s ∘ u) Filter.atTop (𝓝 x)
All goals completed! 🐙
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) := ⊢ IsEquidistributedModuloOne fun n ↦ (3 / 2) ^ n
All goals completed! 🐙
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) := x:ℝhx:Transcendental ℚ x⊢ IsEquidistributedModuloOne fun n ↦ x * (3 / 2) ^ n
All goals completed! 🐙
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 := ⊢ {x | IsAccumulationPoint x fun n ↦ Int.fract ((3 / 2) ^ n)}.Infinite
All goals completed! 🐙
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) := ⊢ IsAccumulationPoint sorry fun n ↦ Int.fract ((3 / 2) ^ n)
All goals completed! 🐙
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)) := ⊢ ∃ p, IsAccumulationPoint p fun n ↦ Int.fract ((3 / 2) ^ n)
All goals completed! 🐙end Equidistribution