**HAL Id: hal-01516246**

**https://hal.sorbonne-universite.fr/hal-01516246**

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**A new result, for the box dimension of the graph of the**

**Weierstrass function**

### Claire David

**To cite this version:**

Claire David. A new result, for the box dimension of the graph of the Weierstrass function. 2017. �hal-01516246�

## A new result, for the box dimension of the graph of the

## Weierstrass function

## Claire David

April 29, 2017Sorbonne Universités, UPMC Univ Paris 06

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France

### 1

### Introduction

The determination of the box and Hausdorﬀ dimension of the graph of the Weierstrass function has,
*since long been, a topic of interest. Let us recall that, given λ* *∈ ]0, 1[, and b such that λ b > 1 +* *3 π*

2 , the Weierstrass function

*x* *∈ R 7→*

+*∞*

∑

*n=0*

*λncos (π bnx)*

is continuous everywhere, while nowhere diﬀerentiable. The original proof, by K. Weierstrass [Wei72],
can also be found in [Tit77]. It has been completed by the one, now a classical one, in the case
*where λ b > 1, by G. Hardy [Har11].*

After the works of A. S. Besicovitch and H. D. Ursell [BU37], it is Benoît Mandelbrot [Man77]
who particularly highlighted the fractal properties of the graph of the Weierstrass function. He also
*conjectured that the Hausdorﬀ dimension of the graph is D _{W}* = 2 +

*ln λ*

*ln b*. Interesting discussions in
relation to this question have been given in the book of K. Falconer [Fal85]. A series of results for
the box dimension can be found in the works of J.-L. Kaplan et al. [KMPY84], where the authors
show that it is equal to the Lyapunov dimension of the equivalent attracting torus, and in those by
T-Y. Hu and K-S. Lau [HL93]. As for the Hausdorﬀ dimension, a proof was given by B. Hunt [Hun98]
in 1998 in the case where arbitrary phases are included in each cosinusoidal term of the summation.
Recently, K. Barańsky, B. Bárány and J. Romanowska [BBR17] proved that, for any value of the real
*number b, there exists a threshold value λb* belonging to the interval

] 1

*b, 1*

[

such that the
*aforemen-tioned dimension is equal to D _{W}*

*for every b in ]λb, 1[. Results by W. Shen [She15] go further than the*

ones of [BBR17]. In [Kel17], G. Keller proposes what appears as a much simpler and very original proof. In our work [Dav17], where we build a Laplacian on the graph of the Weierstrass function, we came across a simpler means of computing the box dimension of the graph, using a sequence a graphs that approximate the studied one. Results are exposed in the sequel.

### 2

### Framework of the study

In this section, we recall results that are developed in [Dav17].

*Notation. In the following, λ and Nb* are two real numbers such that:

*0 < λ < 1* *,* *Nb* *∈ N and λ Nb* *> 1*

We will consider the (1*−periodic) Weierstrass function W, defined, for any real number x, by:*

*W(x) =*

+*∞*

∑

*n=0*

*λn* *cos (2 π N _{b}nx)*

We place ourselves, in the sequel, in the Euclidean plane of dimension 2, referred to a direct
*or-thonormal frame. The usual Cartesian coordinates are (x, y).*

The restriction Γ_{W}*to [0, 1[×R, of the graph of the Weierstrass function, is approximated by means of*
a sequence of graphs, built through an iterative process. To this purpose, we introduce the iterated
*function system of the family of C∞* contractions fromR2 _{to}_{R}2_{:}

*{T*0*, ..., TNb−1}*

*where, for any integer i belonging to{0, ..., Nb− 1}, and any (x, y) of R*2:

*Ti(x, y) =*
(
*x + i*
*Nb*
*, λ y + cos*
(
*2 π*
(
*x + i*
*Nb*
)))
Property 2.1.
Γ* _{W}* =

*N*∪

*b−1*

*i=0*

*Ti*(Γ

*W*)

*Definition 2.1. For any integer i belonging to{0, ..., Nb− 1}, let us denote by:*

*Pi= (xi, yi*) =
(
*i*
*Nb− 1*
*,* 1
1*− λ* cos
(
*2 π i*
*Nb− 1*
))

*the fixed point of the contraction Ti*.

*We will denote by V*0 the ordered set (according to increasing abscissa), of the points:

*{P*0*, ..., PNb−1}*

*The set of points V*0*, where, for any i of{0, ..., Nb− 2}, the point Pi* *is linked to the point Pi+1*,

con-stitutes an oriented graph (according to increasing abscissa)), that we will denote by Γ* _{W}*0

*. V*0 is called the set of vertices of the graph Γ

*0.*

_{W}*For any natural integer m, we set:*

*Vm*=
*N*∪*b−1*

*i=0*

*Ti(Vm−1*)

*The set of points Vm*, where two consecutive points are linked, is an oriented graph (according to

increasing abscissa), which we will denote by Γ* _{W}m. Vm* is called the set of vertices of the graph Γ

*Wm*.

We will denote, in the sequel, by

*NS*

*m* *= 2 Nbm+ Nb− 2*

the number of vertices of the graph Γ* _{W}m*, and we will write:

*Vm*=

{

*Sm*

0 *,S*1*m, . . . ,SNmm−1*

}

Figure 1: The polygons*P1,0*,*P1,1*,*P1,2, in the case where λ =*

1

Figure 2: The graphs Γ_{W}_{0} (in green), Γ_{W}_{1} (in red), Γ_{W}_{2} (in orange), Γ* _{W}* (in cyan), in the case

*where λ =*1

2*, and Nb*= 3.

Definition 2.2. Consecutive vertices on the graph Γ_{W}

*Two points X et Y de Γ _{W}* will be called

*consecutive vertices of the graph Γ*if there exists a

_{W}*natural integer m, and an integer j of{0, ..., Nb− 2}, such that:*

*X = (Ti*1*◦ . . . ◦ Tim) (Pj*) et *Y = (Ti*1 *◦ . . . ◦ Tim) (Pj+1*) *{i*1*, . . . , im} ∈ {0, ..., Nb− 1}*

*m*

or:

*X = (Ti*1 *◦ Ti*2 *◦ . . . ◦ Tim) (PNb−1*) et *Y = (Ti*1+1*◦ Ti*2*. . .◦ Tim) (P*0)

*Definition 2.3. For any natural integer m, the* *N _{m}S* consecutive vertices of the graph Γ

*are, also,*

_{W}m*the vertices of N _{b}m* simple polygons

*Pm,j*, 0

*6 j 6 Nbm− 1, with Nb*

*sides. For any integer j such*

that 0*6 j 6 N _{b}m− 1, one obtains each polygon by linking the point number j to the point *

*num-ber j + 1 if j = i mod Nb*, 0

*6 i 6 Nb− 2, and the point number j to the point number j − Nb*+ 1

*if j =−1 mod Nb*. These polygons generate a Borel set ofR2.

Definition 2.4. Word, on the graph Γ_{W}

*Let m be a strictly positive integer. We will call number-letter any integer* *Mi* of *{0, . . . , Nb− 1},*

and word of length *|M| = m, on the graph Γ _{W}*, any set of number-letters of the form:

*M = (M*1*, . . . ,Mm*)

We will write:

Definition 2.5. Edge relation, on the graph Γ_{W}

*Given a natural integer m, two points X and Y of Γ _{W}m* will be called

*adjacent if and only if X and Y*

are two consecutive vertices of Γ* _{W}m*. We will write:

*X*

*∼*

*m* *Y*

This edge relation ensures the existence of a word *M = (M*1*, . . . ,Mm) of length m, such that X*

*and Y both belong to the iterate:*

*T _{M}V*0

*= (TM*1

*◦ . . . ◦ TMm) V*0

*Given two points X and Y of the graph Γ _{W}, we will say that X and Y are adjacent if and only if*

*there exists a natural integer m such that:*

*X* *∼*

*m* *Y*

Proposition 2.2. *Adresses, on the graph of the Weierstrass function*

*Given a strictly positive integer m, and a word* *M = (M*1*, . . . ,Mm) of length m* *∈ N⋆, on the*

*graph Γ _{W}m, for any integer j of{1, ..., Nb− 2}, any X = TM(Pj) de Vm\ V*0

*, i.e. distinct from one of*

*the Nb*

*fixed point Pi, 06 i 6 Nb− 1, has exactly two adjacent vertices, given by:*

*T _{M}(Pj+1*)

*et*

*TM(Pj−1*)

*where:*

*T _{M}*

*= T*1

_{M}*◦ . . . ◦ TMm*

*By convention, the adjacent vertices of T _{M}(P*0

*) are TM(P*1

*) and TM(PNb−1), those of TM(PNb−1), TM(PNb−2*)

*and T*0

_{M}(P*) .*

*Definition 2.6. mth−order subcell, m ∈ N⋆*, related to a pair of points of the graph Γ_{W}*Given a strictly positive integer m, and two points X and Y of Vm* *such that X∼*

*mY , we will call*

*mth−order subcell, related to the pair of points (X, Y ), the polygon, the vertices of which*

*are X, Y , and the intersection points of the edge between the vertices at the extremities of the polygon,*
i.e. the respective intersection points of polygons of the type *Pm,j−1* and *Pm,j*, 1*6 j 6 Nbm− 1, on*

the one hand, and of the type *Pm,j* and*Pm,j+1*, 0*6 j 6 N _{b}m− 2, on the other hand.*

*Notation. For any integer j belonging to{0, ..., Nb− 1}, any natural integer m, and any word M of*

*length m, we set:*
*T _{M}(Pj*) =

*(x (TM(Pj)) , y (TM(Pj*)))

*,*

*TM(Pj+1*) =

*(x (TM(Pj+1)) , y (TM(Pj+1*)))

*Lm*

*= x (TM(Pj+1*))

*− x (TM(Pj*)) = 1

*(Nb− 1) N*

_{b}m*,*

*hj,m*

*= y (TM(Pj+1*))

*− y (TM(Pj*))

*Figure 3: A mth−order subcell, in the case where λ =* 1

2*, and Nb*= 7.

Proposition 2.3. *An upper bound and lower bound, for the box-dimension of the graph Γ _{W}*

*For any integer j belonging to{0, 1, . . . , Nb− 2}, each natural integer m, and each word M of length m,*

*let us consider the rectangle, the width of which is:*

*Lm* *= x (TM(Pj+1*))*− x (TM(Pj*)) =

1
*(Nb− 1) N _{b}m*

*and height* *|hj,m|, such that the points TM(Pj+1) and TM(Pj+1) are two vertices of this rectangle.*

*Then:*
*L*2*−DW*
*m* *(Nb− 1)*2*−DW*
{_{1}* _{− λ}*2 min
0

*6j6Nb−1*sin (

*π (2 j + 1)*

*Nb− 1*)

*−*

*π*

*Nb(Nb− 1) (λ Nb− 1)*}

*6 |hj,m|*

*and:*

*|hj,m| 6 η*2

*−D*2

_{W}L*m−DW*

*where the real constant η*2*−D _{W}*

*is given by :*

*η*2*−DW* *= 2 π*2*(Nb− 1)*2*−DW*
{
*(2 Nb− 1) λ (N _{b}*2

*− 1)*

*(Nb− 1)*2(1

*− λ) (λ N*2

_{b}*− 1)*+

*2 Nb*

*(λ N*2

_{b}*− 1) (λ N*3

_{b}*− 1)*}

*There exists thus a positive constant*
*C = max*
{
*(Nb− 1)*2*−DW*
{ 2
1*− λ* 0*6j6N*min*b−1*
sin
(
*π (2 j + 1)*
*Nb− 1*
)
*−* *π*
*Nb(Nb− 1) (λ Nb− 1)*
}
*,η*2*−D _{W}*
}

*such that the graph Γ _{W}*

*on Lm*

*can be covered by at least and at most:*

*Nm*
{
*C*
(
*Lm*
*Nm*
)1*−D _{W}*
+ 1
}

*= C L*1

*−DW*

*m*

*NmDW*

*+ Nm*

*squares, the side length of which is* *Lm*
*Nm*

*.*

*Proof. For any pair of integers (im, j) of{0, ..., Nb− 2}*2:

*Tim(Pj*) =
(
*xj+ im*
*Nb*
*, λ yj*+ cos
(
*2 π*
(
*xj+ im*
*Nb*
)))

*For any pair of integers (im, im−1, j) of* *{0, ..., Nb− 2}*3:

*Tim−1(Tim(Pj*)) =
(_{x}_{j}_{+i}_{m}*Nb* *+ im−1*
*Nb*
*, λ*2*yj+ λ cos*
(
*2 π*
(
*xj* *+ im*
*Nb*
))
+ cos
(
*2 π*
(_{x}_{j}_{+i}_{m}*Nb* *+ im−1*
*Nb*
)))
=
(
*xj* *+ im*
*N _{b}*2 +

*i*

_{m−1}*Nb*

*, λ*2

*yj+ λ cos*(

*2 π*(

*xj*

*+ im*

*Nb*)) + cos (

*2 π*(

*xj+ im*

*N*2 +

_{b}*i*

_{m−1}*Nb*)))

*For any pair of integers (im, im−1, im−2, j) of{0, ..., Nb− 2}*4:

*Tim−2*
(
*Tim−1(Tim(Pj*))
)
=
(
*xj+ im*
*N _{b}*3 +

*im−1*

*N*2 +

_{b}*im−2*

*Nb*

*,*

*λ*3

*yj+ λ*2 cos (

*2 π*(

_{x}*j+im*

*Nb*))

*+λ cos*(

*2 π*(

*xj+ im*

*N*2 +

_{b}*im−1*

*Nb*)) + cos (

*2 π*(

*xj+ im*

*N*3 +

_{b}*im−1*

*N*2 +

_{b}*im−2*

*Nb*)) )

*Given a strictly positive integer m, and two points X and Y of Vm* such that:

*X* *∼*

*m* *Y*

there exists a word*M of length |M| = m, on the graph Γ _{W}, and an integer j of{0, ..., Nb− 2}*2, such

that:

*X = T _{M}(Pj*)

*,*

*Y = TM(Pj+1*)

*Let us write T _{M}* under the form:

*T _{M}= Tim◦ Tim−1◦ . . . ◦ Ti*1

*where (i*1

*, . . . , im*)

*∈ {0, ..., Nb− 1}m*.

One has then:

*x (T _{M}(Pj*)) =

*xj*

*N*+

_{b}m*m*∑

*k=1*

*ik*

*N*

_{b}k*,*

*x (TM(Pj+1*)) =

*xj+1*

*N*+

_{b}m*m*∑

*k=1*

*ik*

*N*

_{b}kand:
*y (TM(Pj*)) = *λmyj*+
*m*
∑
*k=1*
*λm−k*cos
(
*2 π*
(
*xj*
*Nk*
*b*
+
*k*
∑
*ℓ=0*
*im−ℓ*
*N _{b}k−ℓ*
))

*y (TM(Pj+1*)) =

*λmyj+1*+

*m*∑

*k=1*

*λm−k*cos (

*2 π*(

*xj+1*

*Nk*

*b*+

*k*∑

*ℓ=0*

*im−ℓ*

*N*))

_{b}k−ℓThis leads to:

*hj,m− λm* *(yj+1− yj*) =
*m*
∑
*k=1*
*λm−k*
{
cos
(
*2 π*
(
*xj+1*
*N _{b}k* +

*k*∑

*ℓ=0*

*i*

_{m−ℓ}*N*))

_{b}k−ℓ*− cos*(

*2 π*(

*xj*

*N*

_{b}k*−*

*k*∑

*ℓ=0*

*i*

_{m−ℓ}*N*))} =

_{b}k−ℓ*−2*

*m*∑

*k=1*

*λm−k*sin (

*π*(

*xj+1− xj*

*N*)) sin (

_{b}k*2 π*(

*xj+1+ xj*

*2 N*+

_{b}k*k*∑

*ℓ=0*

*im−ℓ*

*N*))

_{b}k−ℓTaking into account:

*λm* *(yj+1− yj*) =
*λm*
1*− λ*
(
cos
(
*2 π (j + 1)*
*Nb− 1*
)
*− cos*
(
*2 π j*
*Nb− 1*
))
= *−2* *λ*
*m*
1*− λ* sin
(
*π*
*Nb− 1*
)
sin
(
*π (2 j + 1)*
*Nb− 1*
)
one has:
*hj,m*+ 2
*λm*
1*− λ* sin
(
*π*
*Nb− 1*
)
sin
(
*π (2 j + 1)*
*Nb− 1*
)
= *−2*
*m*
∑
*k=1*
*λm−k*sin
(
*π*
*N _{b}k+1(Nb− 1)*
)
sin
(

*π (2 j + 1)*

*N*

_{b}k+1(Nb− 1)*+ 2 π*

*k*∑

*ℓ=0*

*im−ℓ*

*N*) Thus:

_{b}k−ℓ*y (TM(Pj*))

*− y (TM(Pj+1*))

*− 2*

*λm*1

*− λ*sin (

*π*

*Nb− 1*) sin (

*π (2 j + 1)*

*Nb− 1*) 6 ∑

*m*

*k=1*

*2 π λm−k*

*N*=

_{b}k+1(Nb− 1)*π λm*(

_{1}

*1*

_{−}*λm*

_{N}m*b*)

*λ NbNb(Nb− 1)*( 1

*−*1

_{λ N}*b*) 6

*π λm*

*Nb(Nb− 1) (λ Nb− 1)*

which leads to:

*y (T _{M}(Pj*))

*− y (TM(Pj+1*)) > 2

*λm*1

*− λ*sin (

*π*

*Nb− 1*) sin (

*π (2 j + 1)*

*Nb− 1*)

*−*

*π λm*

*Nb(Nb− 1) (λ Nb− 1)*or:

*y (T _{M}(Pj+1*))

*− y (TM(Pj*)) > 2

*λm*1

*− λ*sin (

*π*

*Nb− 1*) sin (

*π (2 j + 1)*

*Nb− 1*)

*−*

*π λm*

*Nb(Nb− 1) (λ Nb− 1)*

*Due to the symmetric roles played by T _{M}(Pj) and TM(Pj+1*), one may only consider the case when:

*y (T _{M}(Pj*))

*− y (TM(Pj+1*)) > 2

*λm*1

*− λ*sin (

*π*

*Nb− 1*) sin (

*π (2 j + 1)*

*Nb− 1*)

*−*

*π λm*

*Nb(Nb− 1) (λ Nb− 1)*> 0

*> λm*{ 2 1

*− λ*0

*6j6N*min

*b−1*sin (

*π (2 j + 1)*

*Nb− 1*)

*−*

*π*

*Nb(Nb− 1) (λ Nb− 1)*}

The predominant term is thus:

*λm= em (DW−2) ln Nb* _{= N}m (DW−2)

*b* *= L*
2*−D _{W}*

*m* *(Nb− 1)*2*−DW*

One also has:

*|hj,m| 6*
*2 λm*
1*− λ*
*π*2_{(2 j + 1)}*(Nb− 1)*2
+ 2
*m*
∑
*k=1*
*λm−kπ*
{
*2 j + 1*
*(Nb− 1) N _{b}k*
+ 2

*k*∑

*ℓ=0*

*im−ℓ*

*N*}

_{b}k−ℓ*π*

*(Nb− 1) N*=

_{b}k*2 λ*

*m*1

*− λ*

*π*2

*(2 j + 1)*

*(Nb− 1)*2 +

*2 π*2

_{λ}m*Nb− 1*

*m*∑

*k=1*{

*(2 j + 1) λ−k*

*(Nb− 1) Nb2k*+ 2

*k*∑

*ℓ=0*

*i*

_{m−ℓ}λ−k*N*} =

_{b}2k−ℓ*2 λ*

*m*1

*− λ*

*π*2

*(2 j + 1)*

*(Nb− 1)*2 +

*2 π*2

_{λ}m*Nb− 1*{

*λ−1N*

_{b}−2(2 j + 1)*(Nb− 1)*(1

*− λ−mN*) 1

_{b}−2m*− λ−1N*+ 2

_{b}−2*m*∑

*k=1*

*(Nb− 1) λ−k*

*N*1

_{b}2k*− N*1

_{b}−k−1*− N*} 6

_{b}−1*2 λm*1

*− λ*

*π*2

*(2 Nb− 1)*

*(Nb− 1)*2 +

*2 π*2

_{λ}m*Nb− 1*

*(2 Nb− 1)*

*(Nb− 1)*(1

*− λ−mN*)

_{b}−2m*λ N*2

*b*

*− 1*+

*2 π*2

_{λ}m*Nb− 1*2

*λ*

*−1*

_{N}−2*b*

*(Nb− 1) (1 − λ−mNb−2m*) (1

*− N*) (1

_{b}−1*− λ−1N*)

_{b}−2*−2 π*2

*λm*

*Nb− 1*2

*λ*

*−1*

_{N}−3*b*

*(Nb− 1) (1 − λ−mNb−3m*) (1

*− N*) (1

_{b}−1*− λ−1N*) 6

_{b}−3*2 λm*1

*− λ*

*π*2

*(2 Nb− 1)*

*(Nb− 1)*2 +

*2 π*2

_{λ}m*Nb− 1*

*(2 Nb− 1)*

*(Nb− 1)*1

*λ N*2

*b*

*− 1*+

*4 π*2

_{N}*bλm*

*Nb− 1*{ 1

*λ N*2

*b*

*− 1*

*−*1

*λ N*3

*b*

*− 1*} =

*2 π*2

*λm*{

*(2 Nb− 1) λ (N*2

_{b}*− 1)*

*(Nb− 1)*2(1

*− λ) (λ N*2

_{b}*− 1)*+

*2 Nb*

*(λ N*2

_{b}*− 1) (λ N*3

_{b}*− 1)*}

Since:
*x (T _{M}(Pj+1*))

*− x (TM(Pj*)) = 1

*(Nb− 1) Nbm*and:

*D*= 2 +

_{W}*ln λ*

*ln Nb*

*,*

*λ = e(DW−2) ln Nb*

_{= N}(DW−2)*b*

one has thus:

*|hj,m| 6 2 π*2*L*2*m−DW* *(Nb− 1)*2*−DW*
{
*(2 Nb− 1) λ (N _{b}*2

*− 1)*

*(Nb− 1)*2(1

*− λ) (λ N*2

_{b}*− 1)*+

*2 Nb*

*(λ N*2

*b*

*− 1) (λ Nb*3

*− 1)*}

### References

[BBR17] K. Barańsky, B. Bárány, and J. Romanowska. On the dimension of the graph of the classical
*Weierstrass function. Advances in Math., 2017.*

[BU37] *A. S. Besicovitch and H. D. Ursell. Sets of fractional dimensions. Notices of the AMS,*
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