NormalNumber.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Normal numbers

A real number $x$ is simply normal in base $b$ if every digit $d \in \{0,1,\dots,b-1\}$ appears with asymptotic frequency $1/b$ in the base-$b$ expansion of $x$. It is normal in base $b$ if, for every $k \ge 1$, every string of $k$ digits appears with asymptotic frequency $1/b^k$.

A number that is normal in every integer base $b \ge 2$ is called absolutely normal.

Despite extensive empirical evidence, it remains unknown whether classical constants such as $\pi$, $e$, or $\sqrt{2}$ are normal in any base.

References:

Main definitions

digitSeq: the sequence of digits in the base-b fractional expansion of a real number.
IsSimplyNormalInBase: a real number is simply normal in base b.
IsNormalInBase: a real number is normal in base b.
IsAbsolutelyNormal: a real number is normal in every base b ≥ 2.

Imports (2)

Imported by 3 files

def NormalNumber.digitSeq The n-th digit (0-indexed) after the radix point in the base-b expansion of x line 59 source
Source documentation
The n-th digit (0-indexed) after the radix point in the base-b expansion of x.
Concretely, digitSeq b x n = ⌊b^(n+1) * {x}⌋ mod b, where {x} denotes the fractional part of x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HMod.hMod, Nat.floor, HMul.hMul, HPow.hPow, HAdd.hAdd, OfNat.ofNat, Int.fract
19 instances, coercions and auxiliary constants left out.
```
noncomputable def digitSeq (b : ℕ) (x : ℝ) (n : ℕ) : ℕ :=
…
```
def NormalNumber.IsSimplyNormalInBase A real number x is simply normal in base b if every digit d < b appears with asymptotic frequency 1 / b in the base-b fractional expansion of x line 65 source
Source documentation
A real number x is simply normal in base b if every digit d < b appears with asymptotic frequency 1 / b in the base-b fractional expansion of x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: LT.lt, Filter.Tendsto, HDiv.hDiv, Finset.card, Finset.filter, Eq, NormalNumber.digitSeq, Finset.range, Filter.atTop, nhds, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.
```
noncomputable def IsSimplyNormalInBase (b : ℕ) (x : ℝ) : Prop :=
…
```
def NormalNumber.IsNormalInBase Documented declaration: IsNormalInBase line 77 source
Source documentation
A real number x is normal in base b if every string w of k digits < b appears with asymptotic frequency 1 / b ^ k in the base-b fractional expansion of x, that is, the proportion of positions i < n at which w occurs tends to 1 / b ^ k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Fin, LT.lt, Filter.Tendsto, HDiv.hDiv, Finset.card, Finset.filter, Eq, NormalNumber.digitSeq, HAdd.hAdd, Finset.range, Filter.atTop, nhds, OfNat.ofNat, HPow.hPow
21 instances, coercions and auxiliary constants left out.
```
noncomputable def IsNormalInBase (b : ℕ) (x : ℝ) : Prop :=
…
```
theorem NormalNumber.IsNormalInBase.isSimplyNormalInBase A number that is normal in base b is simply normal in base b line 86 source
Source documentation
A number that is normal in base b is simply normal in base b.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, NormalNumber.IsNormalInBase, NormalNumber.IsSimplyNormalInBase
Proof uses: LT.lt, Eq.mpr, Filter.Tendsto, HDiv.hDiv, Finset.card, Finset.filter, Eq, NormalNumber.digitSeq, Finset.range, Filter.atTop, nhds, OfNat.ofNat, Inv.inv, id, congrArg, Filter, one_div, Eq.mp, Fin, HAdd.hAdd and 13 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem IsNormalInBase.isSimplyNormalInBase {b : ℕ} {x : ℝ} (h : IsNormalInBase b x) :
    IsSimplyNormalInBase b x := by
…
```
Mentions: NormalNumber.IsNormalInBase, NormalNumber.IsSimplyNormalInBase
def NormalNumber.IsAbsolutelyNormal A real number x is absolutely normal if it is normal in every integer base b ≥ 2 line 93 source
Source documentation
A real number x is absolutely normal if it is normal in every integer base b ≥ 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Nat, LE.le, OfNat.ofNat, NormalNumber.IsNormalInBase
2 instances, coercions and auxiliary constants left out.
```
noncomputable def IsAbsolutelyNormal (x : ℝ) : Prop :=
…
```