AlgebraicNormality.lean`

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5 declarations, 3 contain sorry, 0 declared as axioms

Normality of Irrational Algebraic Numbers

It is unknown whether every irrational algebraic real number is normal in any integer base. Here a real number is normal in base $b$ if, for every $k \ge 1$, every string of $k$ digits appears in its base-$b$ expansion with asymptotic frequency $1/b^k$. 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.

In a 1950 note on the decimal digits of $\sqrt{2}$, Borel [Bor50] conjectured that every irrational algebraic number is normal in every base. This was the first publication to address the expansion of an irrational algebraic number [AB07].

A real number $x$ is normal in base $b$ if and only if the sequence $(b^n x)_{n \ge 0}$ is uniformly distributed modulo $1$. This was proved by Wall [Wal49]; see also [Bug12, Theorem 4.14] and [EvdPSW03, p. 127].

References:

Wikipedia: Normal number
[Bor50] Borel, Émile. "Sur les chiffres décimaux de $\sqrt{2}$ et divers problèmes de probabilités en chaîne." C. R. Acad. Sci. Paris 230 (1950): 591-593.
[AB07] Adamczewski, Boris, and Yann Bugeaud. "On the complexity of algebraic numbers I. Expansions in integer bases." Annals of Mathematics 165.2 (2007): 547-565.
[Wal49] Wall, Donald Dines. "Normal numbers." Ph.D. thesis, University of California, Berkeley, 1949.
[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Cambridge Tracts in Mathematics 193. Cambridge University Press, 2012.
[EvdPSW03] Everest, Graham, Alf van der Poorten, Igor Shparlinski, and Thomas Ward. "Recurrence sequences." Mathematical Surveys and Monographs 104. American Mathematical Society, Providence, RI, 2003.
[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

Imports (1)

FormalConjecturesUtil

def AlgebraicNormality.IsIrrationalAlgebraic A real number is irrational algebraic if it is algebraic over ℚ but not rational line 62 source
Source documentation
A real number is irrational algebraic if it is algebraic over ℚ but not rational.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: And, IsAlgebraic, Rat, Irrational
5 instances, coercions and auxiliary constants left out.
```
def IsIrrationalAlgebraic (x : ℝ) : Prop :=
…
```
theorem AlgebraicNormality.isNormalInBase_iff_isEquidistributedModuloOne A real number $x$ is normal in base $b$ if and only if $(b^n x)_{n \ge 0}$ is uniformly distributed modulo $1$ [Wal49], [Bug12, Theorem 4.14] sorry line 68 source
Source documentation
A real number $x$ is normal in base $b$ if and only if $(b^n x)_{n \ge 0}$ is uniformly distributed modulo $1$ [Wal49], [Bug12, Theorem 4.14].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, Iff, NormalNumber.IsNormalInBase, IsEquidistributedModuloOne, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isNormalInBase_iff_isEquidistributedModuloOne (b : ℕ) (hb : 2 ≤ b) (x : ℝ) :
    IsNormalInBase b x ↔ IsEquidistributedModuloOne fun n => (b : ℝ) ^ n * x := by
…
```
Mentions: IsEquidistributedModuloOne
theorem AlgebraicNormality.irrational_algebraic_absolutely_normal Borel's normality conjecture [Bor50]: every irrational algebraic real is absolutely normal sorry line 74 source
Source documentation
Borel's normality conjecture [Bor50]: every irrational algebraic real is absolutely normal.

depends on sorry what this means
Statement uses: Iff, True, Real, AlgebraicNormality.IsIrrationalAlgebraic, NormalNumber.IsAbsolutelyNormal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_algebraic_absolutely_normal :
    answer(sorry) ↔ ∀ x : ℝ, IsIrrationalAlgebraic x → IsAbsolutelyNormal x := by
…
```
Mentions: AlgebraicNormality.IsIrrationalAlgebraic
theorem AlgebraicNormality.irrational_algebraic_normal_in_some_base The weaker normality conjecture: every irrational algebraic real is normal in at least one integer base b ≥ 2 sorry line 81 source
Source documentation
The weaker normality conjecture: every irrational algebraic real is normal in at least one integer base b ≥ 2.

depends on sorry what this means
Statement uses: Iff, True, Real, AlgebraicNormality.IsIrrationalAlgebraic, Exists, Nat, And, LE.le, OfNat.ofNat, NormalNumber.IsNormalInBase
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational_algebraic_normal_in_some_base :
    answer(sorry) ↔
      ∀ x : ℝ, IsIrrationalAlgebraic x → ∃ b : ℕ, 2 ≤ b ∧ IsNormalInBase b x := by
…
```
Mentions: AlgebraicNormality.IsIrrationalAlgebraic
theorem AlgebraicNormality.normal_in_some_base_of_absolutely_normal Absolute normality implies normality in at least one base line 88 source
Source documentation
Absolute normality implies normality in at least one base.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, NormalNumber.IsAbsolutelyNormal, Exists, Nat, And, LE.le, OfNat.ofNat, NormalNumber.IsNormalInBase
Proof uses: Exists.intro, And.intro, le_rfl
5 instances, coercions and auxiliary constants left out.
```
theorem normal_in_some_base_of_absolutely_normal {x : ℝ} (hx : IsAbsolutelyNormal x) :
    ∃ b : ℕ, 2 ≤ b ∧ IsNormalInBase b x :=
…
```