1. FormalConjectures
  2. Books
  3. UniformDistributionOfSequences
  4. Equidistribution
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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:

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

def declaration uses 'sorry'isAccumulationPoint_iff_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 declaration uses 'sorry'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 declaration uses 'sorry'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 declaration uses 'sorry'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 declaration uses 'sorry'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 declaration uses 'sorry'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