FormalConjectures.Wikipedia.AlgebraicNormality

FormalConjectures
Wikipedia
AlgebraicNormality

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import FormalConjecturesUtil

Normality of Irrational Algebraic Numbers

It is unknown whether every irrational algebraic real number is normal in any integer base. The stronger conjecture that every irrational algebraic real number is absolutely normal is stated separately: normality in one base and normality in every base are not equivalent definitions.

References:

Wikipedia: Normal number

[BC01] Bailey, David H., and Richard E. Crandall. "On the random character of fundamental constant expansions." Experimental Mathematics 10.2 (2001): 175-190. https://projecteuclid.org/journals/experimental-mathematics/volume-10/issue-2/On-the-random-character-of-fundamental-constant-expansions/em/999188630.full

open NormalNumbernamespace AlgebraicNormality

A real number is irrational algebraic if it is algebraic over ℚ but not rational.

def IsIrrationalAlgebraic (x : ℝ) : Prop :=
  IsAlgebraic ℚ x ∧ Irrational x

The strong normality conjecture: every irrational algebraic real is absolutely normal.

@[category research open, AMS 11 12 41]
theorem irrational_algebraic_absolutely_normal :
    answer(sorry) ↔ ∀ x : ℝ, IsIrrationalAlgebraic x → IsAbsolutelyNormal x := by⊢ True ↔ ∀ (x : ℝ), IsIrrationalAlgebraic x → IsAbsolutelyNormal x
  sorryAll goals completed! 🐙

The weaker normality conjecture: every irrational algebraic real is normal in at least one integer base b ≥ 2.

@[category research open, AMS 11 12 41]
theorem irrational_algebraic_normal_in_some_base :
    answer(sorry) ↔
      ∀ x : ℝ, IsIrrationalAlgebraic x → ∃ b : ℕ, 2 ≤ b ∧ IsNormalInBase b x := by⊢ True ↔ ∀ (x : ℝ), IsIrrationalAlgebraic x → ∃ b, 2 ≤ b ∧ IsNormalInBase b x
  sorryAll goals completed! 🐙

Absolute normality implies normality in at least one base.

@[category API, AMS 11]
theorem normal_in_some_base_of_absolutely_normal {x : ℝ} (hx : IsAbsolutelyNormal x) :
    ∃ b : ℕ, 2 ≤ b ∧ IsNormalInBase b x :=
  ⟨2, le_rfl, hx 2 le_rfl⟩

end AlgebraicNormality