Multifractal analysis of arithmetic functions 10mm @[email protected]

Multifractal analysis of arithmetic functions

Stephane JaffardUniversite Paris Est (France)

Collaborators:

Arnaud Durand Universite Paris Sud Orsay

Samuel Nicolay Universite de Liege

International Conference on Advances on Fractals and Related Topics

Hong-Kong, December 10-14, 2012

Multifractal analysisPurpose of multifractal analysis : Introduce and study classificationparameters for data (functions, measures, distributions, signals,images), which are based on regularity

0.30.2-2

0.0 0.90.1 0.80.60.50.4 1.0

Fonction de Weierstrass : Espace x frequence = 1000x20

Weierstrass function

WH(x) =+∞∑j=0

2−Hj cos(2jx)

0 < H < 1

0.50.4-0.5

0.0 1.00.1 0.30.2 0.90.80.6

Brownien : 1000 points

Brownian motion

0.30.2-2

0.0 0.90.1 0.80.60.50.4 1.0

WH(x) =+∞∑j=0

2−Hj cos(2jx)

0 < H < 1

0.50.4-0.5

0.0 1.00.1 0.30.2 0.90.80.6

Brownian motion

0.30.2-2

0.0 0.90.1 0.80.60.50.4 1.0

WH(x) =+∞∑j=0

2−Hj cos(2jx)

0 < H < 1

0.50.4-0.5

0.0 1.00.1 0.30.2 0.90.80.6

Brownian motion

Everywhere irregular signals and imagesJet turbulence Eulerian velocity signal (ChavarriaBaudetCiliberto95)

0 300 temps (s) 600 9000

70 ∆ = 3.2 ms

Fully developed turbulence Internet Trafic

! "!! #!!! #"!! $!!! $"!! %!!!!!&$

)*+,-./01234

567!689

Euro vs Dollar (2001-2009)

Van Gogh paintingy

300 600 900 1200 1500

Van Gogh paintingy

300 600 900 1200 1500

100 200 300 400 500

Pointwise regularity

Definition :Let f : Rd → R be a locally bounded function and x0 ∈ Rd ;f ∈ Cα(x0) if there exist C > 0 and a polynomial P such that, for|x − x0| small enough,

|f (x)− P(x − x0)| ≤ C|x − x0|α

The Holder exponent of f at x0 is

hf (x0) = sup{α : f ∈ Cα(x0)}

The Holder exponent of the Weierstrass function WH is constant andequal to H (Hardy)

The Holder exponent of Brownian motion is constant and equal to1/2 (Wiener)

WH and B are mono-Holder function

|f (x)− P(x − x0)| ≤ C|x − x0|α

hf (x0) = sup{α : f ∈ Cα(x0)}

|f (x)− P(x − x0)| ≤ C|x − x0|α

hf (x0) = sup{α : f ∈ Cα(x0)}

Multifractal spectrum (Parisi and Frisch, 1985)The iso-Holder sets of f are the sets

EH = {x0 : hf (x0) = H}

Let f be a locally bounded function. The Holder spectrumof f is

Df (H) = dim (EH)

where dim stands for the Hausdorff dimension(by convention, dim (∅) = −∞)

The upper-Holder sets of f are the sets

EH = {x0 : hf (x0) ≥ H}

The lower-Holder sets of f are the sets

EH = {x0 : hf (x0) ≤ H}

EH = {x0 : hf (x0) = H}

Df (H) = dim (EH)

EH = {x0 : hf (x0) ≥ H}

EH = {x0 : hf (x0) ≤ H}

EH = {x0 : hf (x0) = H}

Df (H) = dim (EH)

EH = {x0 : hf (x0) ≥ H}

EH = {x0 : hf (x0) ≤ H}

Riemann’s non-differentiable function and beyond

R2(x) =∞∑

sin(n2x)n2

dF (H) =

4H − 2 if H ∈ [1/2,3/4]0 if H = 3/2

−∞ else

The cubic Riemann function : R3(x) =∞∑

sin(n3x)n3

In a recent paper (arXiv :1208.6533v1) F. Chamizo and A. Ubisconsider

F (x) =∞∑

eiP(n)x

nαdeg(P) = k

Theorem : (Chamizo and Ubis) : let νF be the maximal multiplicityof the zeros of P ′. If 1 + k

2 < α < k and 1k (α− 1) ≤ H ≤ 1

(α− 1

dF (H) ≥ max(νf ,2)(

H − α− 1k

R2(x) =∞∑

sin(n2x)n2

dF (H) =

4H − 2 if H ∈ [1/2,3/4]0 if H = 3/2

−∞ else

sin(n3x)n3

F (x) =∞∑

eiP(n)x

nαdeg(P) = k

2 < α < k and 1k (α− 1) ≤ H ≤ 1

(α− 1

H − α− 1k

R2(x) =∞∑

sin(n2x)n2

dF (H) =

4H − 2 if H ∈ [1/2,3/4]0 if H = 3/2

−∞ else

sin(n3x)n3

F (x) =∞∑

eiP(n)x

nαdeg(P) = k

2 < α < k and 1k (α− 1) ≤ H ≤ 1

(α− 1

H − α− 1k

R2(x) =∞∑

sin(n2x)n2

dF (H) =

4H − 2 if H ∈ [1/2,3/4]0 if H = 3/2

−∞ else

sin(n3x)n3

F (x) =∞∑

eiP(n)x

nαdeg(P) = k

2 < α < k and 1k (α− 1) ≤ H ≤ 1

(α− 1

H − α− 1k

Generalization : Nonharmonic Fourier seriesLet (λn)n∈N be a sequence of points in Rd ; a nonharmonic Fourierseries is a function f that can be written

f (x) =∑

aneiλn·x .

The gap sequence associated with (λn) is the sequence (θn) :

θn = infm 6=n|λn − λm|

The sequence (λn) is separated if : infnθn > 0.

Theorem : Let x0 ∈ Rd . If (λn) is separated and f ∈ Cα(x0), then∃C such that ∀n,

(1) if |λn| ≥ θn, then |an| ≤C

(θn)α.

Thus, ifH = sup{α : (1) holds},

then, for any x0 ∈ Rd , hf (x0) ≤ H.

Open problem : Optimality of this result

f (x) =∑

aneiλn·x .

(θn)α.

f (x) =∑

aneiλn·x .

(θn)α.

f (x) =∑

aneiλn·x .

(θn)α.

f (x) =∑

aneiλn·x .

(θn)α.

Davenport seriesThe sawtooth function is

{x − bxc − 1/2 if x 6∈ Z0 else

rrr0 1

In one variable, Davenport series are of the form

F (x) =∞∑

an{nx}, an ∈ R.

Spectrum estimates for Davenport series

F (x) =∞∑

an{nx}, an ∈ R.

Assuming that (an) ∈ l1, then F is continuous at irrational points andthe jump at p/q (if p ∧ q = 1) is

Bq =∞∑

Theorem : Assume that (nβan) /∈ l∞ and β > 1. Then

dim(EH) ≥Hβ

if (nβan) ∈ l∞and β > 2. Then

dim(EH) ≤2Hβ

Open problem : Sharpen these bounds

F (x) =∞∑

an{nx}, an ∈ R.

Bq =∞∑

dim(EH) ≥Hβ

dim(EH) ≤2Hβ

F (x) =∞∑

an{nx}, an ∈ R.

Bq =∞∑

dim(EH) ≥Hβ

dim(EH) ≤2Hβ

Hecke’s functions

Hs(x) =∞∑

{nx}ns .

The function Hs(x) is a Dirichlet series in the variable s, and itsanalytic continuation depends on Diophantine approximationproperties of x (Hecke, Hardy, Littlewood).

Theorem : If Re(s) ≥ 2, the spectrum of singularities of Hs is

d(H) =2H

Re(s)for H ≤ Re(s)

= −∞ else.

If 1 < Re(s) < 2, the spectrum of singularities of Hecke’s function Hs

satisfiesd(H) =

for H ≤ Re(s)− 1.

Open problem : Improve the second case

Hecke’s functions

Hs(x) =∞∑

{nx}ns .

d(H) =2H

= −∞ else.

satisfiesd(H) =

Hecke’s functions

Hs(x) =∞∑

{nx}ns .

d(H) =2H

= −∞ else.

satisfiesd(H) =

Hecke’s functions (continued)

Hs(x) =∞∑

{nx}ns .

If Re(s) ≤ 1, the sum is no more locally bounded, however :if 1/2 < Re(s) < 1 then Hs ∈ Lp for p < 1

1−β

One can still define a pointwise regularity exponent as follows(Calderon and Zygmund, 1961) :

Definition : Let B(x0, r) denote the open ball centered at x0 and ofradius r ; α > −d/p. Let f ∈ Lp. Then f belongs to T p

α(x0) if∃C,R > 0 and a polynomial P such that

∀r ≤ R,

∫B(x0,r)

|f (x)− P(x − x0)|pdx

≤ Crα.

The p-exponent of f at x0 is : hpf (x0) = sup{α : f ∈ T p

α(x0)}.The p-spectrum of f is : dp

f (H) = dim({x0 : hp

f (x0) = H})

Open problem : Determine the p-spectrum of Hecke’s functions

The Lebesgue-Davenport functionLet t ∈ [0,1) andt = (0; t1, t2, . . . , tn, . . . )2

be its proper expansion in basis 2.

Then L(t) = (x3(t), y3(t)) where{x3(t) = (0; t1, t3, t5, . . . )2y3(t) = (0; t2, t4, t6, . . . )2.

The Lebesgue-Davenport function L has the following expansion

x3(t) =12+∑

an{2nt} where a2n = 2−n and a2n+1 = −2−n−1

y3(t) =12+∑

bn{2nt} where b2n = −2−n and b2n+1 = 2−n.

The spectrum of singularities of L is{dL(H) = 2H if 0 ≤ H ≤ 1/2

= −∞ else.

The Lebesgue-Davenport function L has the following expansion

x3(t) =12+∑

an{2nt} where a2n = 2−n and a2n+1 = −2−n−1

y3(t) =12+∑

bn{2nt} where b2n = −2−n and b2n+1 = 2−n.

The spectrum of singularities of L is{dL(H) = 2H if 0 ≤ H ≤ 1/2

= −∞ else.

Davenport series in several variablesDavenport series in several variables are of the form

f (x) =∑n∈Zd

an{n · x}

where (an)n∈Zd is an odd sequence indexed by Zd .

Discontinuities of Davenport seriesFor p ∈ Z and q ∈ Zd

∗ , let

Hp,q = {x ∈ Rd | p = q · x}

Let us assume that (an)n∈Zd is an odd sequence in `1. Then,The Davenport series is continuous except on the set

⋃Hp,q where it

has a jump of magnitude |Aq | with

Aq = 2∞∑l=1

Davenport series in several variablesDavenport series in several variables are of the form

f (x) =∑n∈Zd

an{n · x}

where (an)n∈Zd is an odd sequence indexed by Zd .

Discontinuities of Davenport seriesFor p ∈ Z and q ∈ Zd

∗ , let

Hp,q = {x ∈ Rd | p = q · x}

Let us assume that (an)n∈Zd is an odd sequence in `1. Then,The Davenport series is continuous except on the set

⋃Hp,q where it

has a jump of magnitude |Aq | with

Aq = 2∞∑l=1

Upper bound on the Holder exponent of a Davenportseries

For each q ∈ Zd , let Pq = {p ∈ Z | gcd(p,q) = 1}.For x0 ∈ Rd , let

δPq (x0) = dist

x0,⋃

p∈Pq

Let f be a Davenport series with jump sizes (Aq)q∈Zd . Then,

∀x0 ∈ Rd hf (x0) ≤ lim infq→∞Aq 6=0

log |Aq |log δPq (x0)

Connection with Diophantine approximation :

|q · x0 − p| < |q| |Aq |1/α for an infinite sequence =⇒ hf (x0) ≤ α.

Corollary : If the jumps Aq satisfy : |Aq | ≥ C/qa for all q in onedirection at least, then

∀x , hf (x) ≤ a/2 and d(EH) ≤ d − 1 +2Ha

Upper bound on the Holder exponent of a Davenportseries

For each q ∈ Zd , let Pq = {p ∈ Z | gcd(p,q) = 1}.For x0 ∈ Rd , let

δPq (x0) = dist

x0,⋃

p∈Pq

Let f be a Davenport series with jump sizes (Aq)q∈Zd . Then,

∀x0 ∈ Rd hf (x0) ≤ lim infq→∞Aq 6=0

log |Aq |log δPq (x0)

Connection with Diophantine approximation :

|q · x0 − p| < |q| |Aq |1/α for an infinite sequence =⇒ hf (x0) ≤ α.

Corollary : If the jumps Aq satisfy : |Aq | ≥ C/qa for all q in onedirection at least, then

∀x , hf (x) ≤ a/2 and d(EH) ≤ d − 1 +2Ha

Sparse Davenport seriesA Davenport series with coefficients given by a sequence (an)n∈Zd issparse if

limR→∞

log#{|n| < R | an 6= 0}log R

A sequence a = (an)n∈Zd belongs to Fγ if

|an| ≤C|n|γ

Theorem : Let f be a Davenport series with coefficientsa = (an)n∈Zd . Let

γa := sup{γ > 0 | (an)n∈Zd ∈ Fγ}

We assume that f is sparse and that 0 < γa <∞. Then,

∀H ∈ [0, γa] df (H) = d − 1 +Hγa,

else df (H) = −∞

limR→∞

log#{|n| < R | an 6= 0}log R

|an| ≤C|n|γ

∀H ∈ [0, γa] df (H) = d − 1 +Hγa,

limR→∞

log#{|n| < R | an 6= 0}log R

|an| ≤C|n|γ

∀H ∈ [0, γa] df (H) = d − 1 +Hγa,

Open problems concerning multivariate Davenportseries

I Understand when the upper bound for the Holder exponent issharp

I Mutivariate analogue of Hecke’s function

Hs(x) =∑ {n · x}

where the sum is taken on an half-plane

I What can be the shape of the spectrum of singularities of aDavenport series ?

I Directional regularity

Thank you for your (fractal ?) attention !

Hs(x) =∑ {n · x}

Multifractal analysis of arithmetic functions 10mm @[email protected]

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