ExpansionEntropy.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Entropy of the expansions of a real number

The entropy of the continued fraction expansion of a real number $\xi$ and of its expansion in an integer base $b$. Both are the entropy SymbolicDynamics.blockEntropy of the sequence of digits.

References:

[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.

Main definitions

Real.partQuot: the sequence of partial quotients of a real number.
Real.cfEntropy: the entropy of the continued fraction expansion of a real number.
Real.baseEntropy: the entropy of the base b expansion of a real number.

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Real.partQuot The sequence $(c_n)_{n \ge 1}$ of partial quotients of the continued fraction expansion $\xi = [c_0; c_1, c_2, \ldots]$, indexed from $0$ line 52 source
Source documentation
The sequence $(c_n)_{n \ge 1}$ of partial quotients of the continued fraction expansion $\xi = [c_0; c_1, c_2, \ldots]$, indexed from $0$. The integer part $c_0$ is not part of the sequence. The expansion of an irrational number never terminates, so the default value 0 is never used for such $\xi$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: Option.getD, Stream'.Seq.get?, GenContFract.partDens, GenContFract.of, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
noncomputable def partQuot (ξ : ℝ) (n : ℕ) : ℝ :=
…
```
def Real.cfEntropy The entropy $E(\xi)$ of the continued fraction expansion of $\xi$ line 56 source
Source documentation
The entropy $E(\xi)$ of the continued fraction expansion of $\xi$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, EReal
Proof uses: SymbolicDynamics.blockEntropy, Real.partQuot
```
noncomputable def cfEntropy (ξ : ℝ) : EReal := blockEntropy (partQuot ξ)
```
Mentions: Real.partQuot
def Real.baseEntropy The entropy $E(\xi, b)$ of the base $b$ expansion of $\xi$ line 59 source
Source documentation
The entropy $E(\xi, b)$ of the base $b$ expansion of $\xi$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, EReal
Proof uses: SymbolicDynamics.blockEntropy, NormalNumber.digitSeq
```
noncomputable def baseEntropy (b : ℕ) (ξ : ℝ) : EReal :=
…
```