Bernoulli-type measures and the greedy beta-expansion

Tuomas Sahlsten

Winter School on Multifractals and Number TheoryUniversity of Bremen, 22.3.2013

Joint work with 李兵

Sidorov’s TheoremIf 1 < β < 2, then Lebesgue a.e. x ∈ [0, 1

β−1 ] has continuum β-expansions.

Sidorov’s TheoremIf 1 < β < 2, then Lebesgue a.e. x ∈ [0, 1

β−1 ] has continuum β-expansions.

Sidorov’s TheoremIf 1 < β < 2, then Lebesgue a.e. x ∈ [0, 1

β−1 ] has continuum β-expansions.

Sidorov’s TheoremIf 1 < β < 2, then Lebesgue a.e. x ∈ [0, 1

β−1 ] has continuum β-expansions.

Sidorov’s TheoremIf 1 < β < 2, then Lebesgue a.e. x ∈ [0, 1

β−1 ] has continuum β-expansions.

Greedy tree

Let Σβ ⊂ {0, 1}N be the subtree of greedy expansions of numbers in [0, 1).

• The (greedy) β-shift is the dynamical system (Σβ, σβ); σβ = σ|Σβ

• Example: if β is the Golden ratio 1+√5

2 , then

Σβ = {w ∈ {0, 1}N : 11 /∈ w}

Greedy tree

Let Σβ ⊂ {0, 1}N be the subtree of greedy expansions of numbers in [0, 1).

• The (greedy) β-shift is the dynamical system (Σβ, σβ); σβ = σ|Σβ

• Example: if β is the Golden ratio 1+√5

2 , then

Σβ = {w ∈ {0, 1}N : 11 /∈ w}

Greedy tree

Let Σβ ⊂ {0, 1}N be the subtree of greedy expansions of numbers in [0, 1).

• The (greedy) β-shift is the dynamical system (Σβ, σβ); σβ = σ|Σβ

• Example: if β is the Golden ratio 1+√5

2 , then

Σβ = {w ∈ {0, 1}N : 11 /∈ w}

Greedy tree

Let Σβ ⊂ {0, 1}N be the subtree of greedy expansions of numbers in [0, 1).

• The (greedy) β-shift is the dynamical system (Σβ, σβ); σβ = σ|Σβ

• Example: if β is the Golden ratio 1+√5

2 , then

Σβ = {w ∈ {0, 1}N : 11 /∈ w}

Greedy tree

Let Σβ ⊂ {0, 1}N be the subtree of greedy expansions of numbers in [0, 1).

• The (greedy) β-shift is the dynamical system (Σβ, σβ); σβ = σ|Σβ

• Example: if β is the Golden ratio 1+√5

2 , then

Σβ = {w ∈ {0, 1}N : 11 /∈ w}

Aim

Besicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

AimBesicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

AimBesicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

AimBesicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

AimBesicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

AimBesicovitch-Eggleston (multifractal) analysis for (greedy) β-expansions?

E.g. the geometry of

Fp,β = πβ

{w ∈ Σβ :

1

n

n−1∑k=0

1(wk = 0)n→∞−→ p

}?

Here πβ : {0, 1}N → [0, 1β−1 ] is the projection πβ(w) =

∑wiβ

−i.

Theorem (Pfister, Sullivan 2005)

dimHFp,β = sup{hµ(σβ)

log β: σβµ = µ, µ[0] = p

}

• If (Σβ, σβ) is of finite type (i.e. 1 = ε1β−1 + · · ·+ εMβ

−M ), we havethermodynamical formalism at our disposal (Fan, Feng, Wu ’01)...

...so we are done? What if (Σβ, σβ) is not SFT?

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Can we follow the ideas of Besicovitch-Eggleston?

First step. Find a “Bernoulli measure” on the symbolic space Σβ .

• A ’Bernoulli-type’ measure µp:

Properties of µp

• We are missing the precious independence/product structure, i.e.

σβµp 6= µp and sometimes µp[ww′] 6= µp[w]µp[w

′]...

...so no strong law of large numbers (ergodic theorem) for us /

• However, µp is ‘quasi-invariant’ in the sense that

µp � σβµp � µp

and also satisfies the ’ergodic property’

A = σ−1β A =⇒ µp(A) = 0 or 1...

...but these alone do not give us strong enough ergodic theory

Properties of µp

• We are missing the precious independence/product structure, i.e.

σβµp 6= µp and sometimes µp[ww′] 6= µp[w]µp[w

′]...

...so no strong law of large numbers (ergodic theorem) for us /

• However, µp is ‘quasi-invariant’ in the sense that

µp � σβµp � µp

and also satisfies the ’ergodic property’

A = σ−1β A =⇒ µp(A) = 0 or 1...

...but these alone do not give us strong enough ergodic theory

Properties of µp

• We are missing the precious independence/product structure, i.e.

σβµp 6= µp and sometimes µp[ww′] 6= µp[w]µp[w

′]...

...so no strong law of large numbers (ergodic theorem) for us /

• However, µp is ‘quasi-invariant’ in the sense that

µp � σβµp � µp

and also satisfies the ’ergodic property’

A = σ−1β A =⇒ µp(A) = 0 or 1...

...but these alone do not give us strong enough ergodic theory

What is ‘enough’ then?

Quasi-product measures

A measure µ on Σβ is quasi-product (or quasi-Bernoulli), if there existsC ≥ 1 s.t.

C−1µ[w]µ[w′] ≤ µ[ww′] ≤ Cµ[w]µ[w′]

for w,w′ ∈ Σ∗β such that ww′ ∈ Σ∗β .

Unfortunately, this is too strong requirement for measures µp in general:

Theorem (Li, S. 2013)

Let 1 < β < 2 and 0 < p < 1. Then

(Σβ, σβ) is a subshift of finite type ⇐⇒ µp is quasi-product

What is ‘enough’ then?

Quasi-product measures

A measure µ on Σβ is quasi-product (or quasi-Bernoulli), if there existsC ≥ 1 s.t.

C−1µ[w]µ[w′] ≤ µ[ww′] ≤ Cµ[w]µ[w′]

for w,w′ ∈ Σ∗β such that ww′ ∈ Σ∗β .

Unfortunately, this is too strong requirement for measures µp in general:

Theorem (Li, S. 2013)

Let 1 < β < 2 and 0 < p < 1. Then

(Σβ, σβ) is a subshift of finite type ⇐⇒ µp is quasi-product

What is ‘enough’ then?

Quasi-product measures

A measure µ on Σβ is quasi-product (or quasi-Bernoulli), if there existsC ≥ 1 s.t.

C−1µ[w]µ[w′] ≤ µ[ww′] ≤ Cµ[w]µ[w′]

for w,w′ ∈ Σ∗β such that ww′ ∈ Σ∗β .

Unfortunately, this is too strong requirement for measures µp in general:

Theorem (Li, S. 2013)

Let 1 < β < 2 and 0 < p < 1. Then

(Σβ, σβ) is a subshift of finite type ⇐⇒ µp is quasi-product

What is ‘enough’ then?

Quasi-product measures

A measure µ on Σβ is quasi-product (or quasi-Bernoulli), if there existsC ≥ 1 s.t.

C−1µ[w]µ[w′] ≤ µ[ww′] ≤ Cµ[w]µ[w′]

for w,w′ ∈ Σ∗β such that ww′ ∈ Σ∗β .

Unfortunately, this is too strong requirement for measures µp in general:

Theorem (Li, S. 2013)

Let 1 < β < 2 and 0 < p < 1. Then

(Σβ, σβ) is a subshift of finite type ⇐⇒ µp is quasi-product

Sketch of the proof I

• Numbers for tracking branching:

N0(w) = ]{1 ≤ k ≤ |w| : wk = 0 and w1 . . . wk−11 ∈ Σ∗β};

N1(w) = ]{1 ≤ k ≤ |w| : wk = 1}.

• By definitionµp[w] = pN0(w)(1− p)N1(w)...

...and no matter how we choose β, we have

N0(ww′) ≤ N0(w) +N0(w

′) and N1(ww′) = N1(w) +N1(w

′).

• Thus

µp[ww′] � µp[w]µp[w

′] ⇐⇒ there exists M ≥ 0 withN0(ww

′) ≥ N0(w) +N0(w′)−M.

Sketch of the proof I

• Numbers for tracking branching:

N0(w) = ]{1 ≤ k ≤ |w| : wk = 0 and w1 . . . wk−11 ∈ Σ∗β};

N1(w) = ]{1 ≤ k ≤ |w| : wk = 1}.

• By definitionµp[w] = pN0(w)(1− p)N1(w)...

...and no matter how we choose β, we have

N0(ww′) ≤ N0(w) +N0(w

′) and N1(ww′) = N1(w) +N1(w

′).

• Thus

µp[ww′] � µp[w]µp[w

′] ⇐⇒ there exists M ≥ 0 withN0(ww

′) ≥ N0(w) +N0(w′)−M.

Sketch of the proof I

• Numbers for tracking branching:

N0(w) = ]{1 ≤ k ≤ |w| : wk = 0 and w1 . . . wk−11 ∈ Σ∗β};

N1(w) = ]{1 ≤ k ≤ |w| : wk = 1}.

• By definitionµp[w] = pN0(w)(1− p)N1(w)...

...and no matter how we choose β, we have

N0(ww′) ≤ N0(w) +N0(w

′) and N1(ww′) = N1(w) +N1(w

′).

• Thus

µp[ww′] � µp[w]µp[w

′] ⇐⇒ there exists M ≥ 0 withN0(ww

′) ≥ N0(w) +N0(w′)−M.

Sketch of the proof I

• Numbers for tracking branching:

N0(w) = ]{1 ≤ k ≤ |w| : wk = 0 and w1 . . . wk−11 ∈ Σ∗β};

N1(w) = ]{1 ≤ k ≤ |w| : wk = 1}.

• By definitionµp[w] = pN0(w)(1− p)N1(w)...

...and no matter how we choose β, we have

N0(ww′) ≤ N0(w) +N0(w

′) and N1(ww′) = N1(w) +N1(w

′).

• Thus

µp[ww′] � µp[w]µp[w

′] ⇐⇒ there exists M ≥ 0 withN0(ww

′) ≥ N0(w) +N0(w′)−M.

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . .

...so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

Sketch of the proof II⇐ If (Σβ, σβ) is an M -step SFT, i.e. 1 = ε1β

−1 + · · ·+ εMβ−M ,

N0(ww′) ≥ N0(w) +N0(w

′)−M.

Hence µp is quasi-product.

Lemma (Growth of N0)

For any 1 < β < 2 and w ∈ Σβ , πβ(w) 6= 1, we have

N0(w1w2 . . . wk)→∞, as k →∞.

⇒ If (Σβ, σβ) is not SFT, then 1 has infinite β-expansion ε1ε2ε3 . . ....so we can choose kn →∞ with

N0(ε2 . . . εkn) > n, n ∈ N

=⇒ N0(ε1)︸ ︷︷ ︸=0

+ N0(ε2 . . . εkn)− n > N0(ε1ε2 . . . εkn)︸ ︷︷ ︸=0

, n ∈ N.

Hence µp is not quasi-product. ‘�’

The next step?

• (Σβ, σβ) is SFT ⇐⇒ there exists a unique σβ invariant ergodicmeasure µ′p equivalent to µp? (this is what we actually fear)• What is the structure of µp measures when using ‘lazier’ β-expansions

to construct the tree Σβ?• When is the projection πβµp abs. cont. to Lebesgue measure on [0, 1]?

Conjecture (A.-H. Fan 2012, personal discussion):(a) ∀β ∈ (1, 2) ∃ unique p ∈ (0, 1) s.t. πβµp � L,(b) ∀p ∈ (0, 1) ∃ unique β ∈ (1, 2) s.t. πβµp � L.

The next step?

• (Σβ, σβ) is SFT ⇐⇒ there exists a unique σβ invariant ergodicmeasure µ′p equivalent to µp? (this is what we actually fear)

• What is the structure of µp measures when using ‘lazier’ β-expansionsto construct the tree Σβ?• When is the projection πβµp abs. cont. to Lebesgue measure on [0, 1]?

Conjecture (A.-H. Fan 2012, personal discussion):(a) ∀β ∈ (1, 2) ∃ unique p ∈ (0, 1) s.t. πβµp � L,(b) ∀p ∈ (0, 1) ∃ unique β ∈ (1, 2) s.t. πβµp � L.

The next step?

• (Σβ, σβ) is SFT ⇐⇒ there exists a unique σβ invariant ergodicmeasure µ′p equivalent to µp? (this is what we actually fear)• What is the structure of µp measures when using ‘lazier’ β-expansions

to construct the tree Σβ?

• When is the projection πβµp abs. cont. to Lebesgue measure on [0, 1]?

Conjecture (A.-H. Fan 2012, personal discussion):(a) ∀β ∈ (1, 2) ∃ unique p ∈ (0, 1) s.t. πβµp � L,(b) ∀p ∈ (0, 1) ∃ unique β ∈ (1, 2) s.t. πβµp � L.

The next step?

• (Σβ, σβ) is SFT ⇐⇒ there exists a unique σβ invariant ergodicmeasure µ′p equivalent to µp? (this is what we actually fear)• What is the structure of µp measures when using ‘lazier’ β-expansions

to construct the tree Σβ?• When is the projection πβµp abs. cont. to Lebesgue measure on [0, 1]?

Conjecture (A.-H. Fan 2012, personal discussion):(a) ∀β ∈ (1, 2) ∃ unique p ∈ (0, 1) s.t. πβµp � L,(b) ∀p ∈ (0, 1) ∃ unique β ∈ (1, 2) s.t. πβµp � L.

The next step?

• (Σβ, σβ) is SFT ⇐⇒ there exists a unique σβ invariant ergodicmeasure µ′p equivalent to µp? (this is what we actually fear)• What is the structure of µp measures when using ‘lazier’ β-expansions

to construct the tree Σβ?• When is the projection πβµp abs. cont. to Lebesgue measure on [0, 1]?

Conjecture (A.-H. Fan 2012, personal discussion):(a) ∀β ∈ (1, 2) ∃ unique p ∈ (0, 1) s.t. πβµp � L,(b) ∀p ∈ (0, 1) ∃ unique β ∈ (1, 2) s.t. πβµp � L.

Bernoulli convolution

Bernoulli convolution

Bernoulli convolution

Bernoulli convolution

Bernoulli convolution