Examples of recurrent or transient stationary walks in $\RR^d$ over a rotation of $\TT^2$

HAL Id: hal-00456584

https://hal.archives-ouvertes.fr/hal-00456584

Submitted on 22 Mar 2018

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Examples of recurrent or transient stationary walks in ^d over a rotation of ²

Nicolas Chevallier, Jean-Pierre Conze

To cite this version:

Nicolas Chevallier, Jean-Pierre Conze. Examples of recurrent or transient stationary walks in

^d

over

a rotation of

²

. Idris Assani. Ergodic theory: Papers from the Probability and Ergodic Theory

Workshops held at the University of North Carolina, American Mathematical Society, pp.71-84, 2009,

Contemporary Mathematics n° 485. �hal-00456584�

R ^d T ²

R

^d

T

²

(E, A , µ) τ E

µ ϕ E R

^d

ϕ (E, µ, τ ) R

^d

(ϕ, τ ) (ϕ

n

)

n∈N

ϕ

n

:= !

_n−1

k=0

ϕ ◦ τ

^k

n ≥ 0 τ

ϕ

E × R

^d

τ

ϕ

: (x, y) → (τ x, ϕ(x) + y).

τ

ϕ

λ := µ × m m

R

^d

(ϕ

n

) R

^d

(E, µ, τ ) x (ϕ

n

(x))

_n≥0

0 τ

ϕ

λ

ϕ R

(E, τ, µ) (ϕ

_n

)

_n∈N

µ(ϕ) = 0 ϕ

R

^d

d > 1

R

^d

T

²

% %

²

| | R

^d

(ϕ, τ ) (E, A , µ, τ )

R

^d

(k

n

)

(δ

_n

> 0)

lim

n

µ(x : | ϕ

kn

(x) | ≥ δ

n

) = 0 and δ

n

= o(n

^1/d

), (ϕ, τ )

τ ( ˜ E, A ˜ , µ, ˜ τ ˜ )

(ϕ, τ ) ( ˜ ϕ, τ ˜ ) µ ˜

µ τ

B τ

ϕ

(τ

_ϕ^"

B, % ≥ 0)

λ(B) = 0 λ(B) > 0 B

E × K K R

^d

E

n

= { x : | ϕ

kn

(x) | ≤ δ

n

} B

n

= (E

n

× K) "

B n

0

λ(B

j

) ≥

¹₂

λ(B) j ≥ n

0

λ( ∪

ⁿj=n0

τ

ϕ^kj

B

j

) ≥

¹₂

(n − n

0

)λ(B)

ρ

n

:= sup

_1≤k≤n

δ

k

(ρ

n

) (δ

n

) ρ

n

=

o(n

^1/d

)

∪

ⁿj=n0

τ

ϕ^kj

B

_j

E × (D(ρ

_n

) + K) D(ρ

_n

) ρ

n

C

λ( ∪

ⁿj=n0

τ

_ϕ^kj

B

j

) ≤ Cm(D(ρ

n

)).

lim inf

n

m(D(ρ

n

))

n ≥ 1

2C λ(B) > 0 ρ

_n

= o(n

^1/d

) !

% ϕ

n

%

²

= o(n

^1d

)

E

n

:= { x : | ϕ

n

(x) | ≤ δ

n

} (δ

n

) > 0 µ(E

_n^c

) ≤ (δ

_n⁻¹

% ϕ

n

%

²

)

²

δ

_n⁻¹

% ϕ

_n

%

2

→ 0, n

^−1/d

δ

_n

→ 0.

δ

n

:= (n

^1/d

% ϕ

n

%

²

)

^1/2

!

(E, A , µ, τ )

(ϕ, τ ) R

^d

d T

^r

r > 1

C R

^d

T

^r

T

²

m

Z

²_∗

= Z

²

\{ 0 }

x | x |

0

:= max(1, | x | ) || x || := inf( { x } , { 1 − x } ) = inf

n∈Z

| x − n | R

0

: R

²

→ R R

0

(h

1

, h

2

) := | h

1

|

₀

| h

2

|

₀

%

1

%

2

R

²

R

"1,"2

: R

²

→ R

R

_"1,"2

(h) := | %

₁

(h) |

0

| %

₂

(h) |

0

.

∆ T

²

1

∆

∆ ϕ

∆

:= 1

∆

− m(∆)

C %

1

, %

2

, ...%

p

h ∈ Z

²

| ϕ #

∆

(h) | ≤ C $

1≤i<j≤p

1 R

_"i,"j

(h) .

∆ ∆

₁

, ..., ∆

_n

T

1

, ..., T

n

[0, 1]

²

∆

k

T

_k

T = { (x, y) ∈ [0, 1]

²

: x ≤ y }

1 %

T

(ξ

1

, ξ

2

) = 1 2iπξ

1

( 1 − e

^−2iπξ2

2iπξ

2

− 1 − e

^{−2iπ(ξ1+ξ2)}

2iπ(ξ

1

+ ξ

2

) )

&

&

&% 1

T

(ξ

1

, ξ

2

) & & & ≤ C( 1

| ξ

1

|

0

| ξ

2

|

0

+ 1

| ξ

1

|

0

| ξ

1

+ ξ

2

|

0

+ 1

| ξ

2

|

0

| ξ

1

+ ξ

2

|

0

). !

%

1

%

2

R

²

C

1

, C

2

δ > 0

$

h∈Z²_∗

1

R

"1,"2

(h)

^1+δ

∈ [C

1

δ

⁻²

, C

2

δ

⁻²

].

L = (%

1

, %

2

) ! Λ Z

²

L

h∈Λ 1 R0(h)^1+δ

Λ x + [0, 1]

²

M x ∈ R

²

C

2

$

h∈Λ

1

R

0

(h)

^1+δ

≤ M $

h∈Z²

1

R

0

(h)

^1+δ

< C

2

δ

⁻²

. C

1

δ

⁻²

!

%

1

, %

2

R

²

α ∈ T

²

δ > 0

K

δ

(α) := sup

h∈Z²_∗

{ R

"1,"2

(h)

^−(1+δ)

% h.α %

⁻¹

} h ∈ Z

²

ε > 0 A

h,δ,ε

:= { α ∈ T

²

: % h.α % ≤

_R" ^ε

1,"2(h)^1+δ

}

ε > 0 α ∈ T

²

c > 0

% k.α % ≥ c | k |

^−(2+ε)

, ∀ k ∈ Z

²_∗

. K

δ

α K

δ

(α) < + ∞ α

K

δ

R R

l1,l2

δ > 0 m( '

h∈Z²_∗

A

h,δ,ε

) ≥ 1

3 min(1, δ 2

$

h∈Z²_∗

m(A

h,δ,ε

)).

N ≥ 1 Λ

d

h = (h

1

, h

2

)

gcd(h

1

, h

2

) = d Λ

d

= dΛ

1

d

$

0<|h|≤N

m(A

h,δ,ε

) = $

d≥1

$

h∈Λd,0<|h|≤N

m(A

h,δ,ε

) = $

d≥1

$

h∈dΛ1,0<|h|≤N

2ε R(h)

^1+δ

≤ $

d≥1

$

h∈Λ1,0<|h|≤N

2ε

R(dh)

^1+δ

≤ $

h∈Λ1,0<|h|≤N

$

d≥1

2ε d

^1+δ

R(h)

^1+δ

≤ 2 δ

$

h∈Λ1,0<|h|≤N

2ε

R(h)

^1+δ

= 2 δ

$

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

).

A

^(N)

= '

h∈Λ1,0<|h|≤N

A

_h,δ,ε

.

(

A^(N)

$

h∈Λ1,0<|h|≤N

1

Ah,δ,ε

dm ≤ (m(A

^(N)

))

^1/2

( (

A^(N)

( $

h∈Λ1,0<|h|≤N

1

Ah,δ,ε

)

²

dm)

^1/2

,

m(A

^(N)

) ≥ ( !

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

))

²

D ,

D = $

h&=k∈Λ1,0<|h|,|k|≤N

m(A

h,δ,ε

∩ A

k,δ,ε

) + $

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

).

h k Λ

1

h = ± k h k

Z

²

a b

m( { α : % h.α % ≤ a } ∩ { α : % k.α % ≤ b } ) = m( { α : % h.α % ≤ a } ) m( { α : % k.α % ≤ b } ).

$

h&=k∈Λ1,0<|h|,|k|≤N

m(A

h,δ,ε

∩ A

k,δ,ε

) = $

h&=±k∈Λ1,0<|h|,|k|≤N

m(A

h,δ,ε

) m(A

k,δ,ε

)+ $

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

),

D ≤ ( $

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

))

²

+ 2 $

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

).

v

²

v

²

+ 2v ≥ 1

3 min(1, v), ∀ v > 0 m(A

^(N)

) ≥ 1

3 min(1, $

h∈Λ1,0<|h|≤N

m(A

h,δ,ε

))

≥ 1

3 min(1, δ 2

$

0<|h|≤N

m(A

h,δ,ε

)).

N + ∞ !

K

δ

L

^s

[0, 1] s < 1

C C

^'

δ > 0 s ∈ ]0, 1[

C δ

^s

s 1 − s ≤

(

T²

K

δ

(α)

^s

dα ≤ 1 + C

^'

δ

²

1 1 − s . σ

δ

:= !

h∈Z²_∗

R(h)

^−(1+δ)

0 < C

1

< C

2

σ

_δ

δ

²

∈ [C

₁

, C

₂

] s ∈ ]0, 1[

t > 0 A

^s_t

:= { α ∈ T

²

| K

δ

(α)

^s

> t } '

h&=0

A

h,δ,t^−1s

m(A

h,δ,t^−1s

) = min(1, 2R(h)

^−(1+δ)

t

^−1s

) (

T²

K

δ

(α)

^s

dα = (

_∞

0

m(A

^s_t

) dt ≤ 1 + 2 (

_+∞

1

$

h∈Z²_∗

R(h)

^−(1+δ)

t

^−1s

dt

< 1 + 2 1 − s

$

h∈Z²_∗

R(h)

^−(1+δ)

≤ 1 + 1 1 − s

C

2

δ

²

.

(

T²

K

δ

(α)

^s

dα = (

_∞

0

m(A

^s_t

) dt ≥ (

_∞

0

1

3 min(1, δ 2

$

h∈Z²_∗

min(1, 2R(h)

^−(1+δ)

t

^−1/s

) dt

≥ δ 3

(

_∞

0

$

h∈Z²_∗

min( 1 δσ

δ

R(h)

^−(1+δ)

, 1

2 , R(h)

^−(1+δ)

t

^−1/s

)dt

≥ δ 3

$

h∈Z²_∗

(

_∞

0

min( 1 δσ

δ

R(h)

^−(1+δ)

, R(h)

^−(1+δ)

t

^−1/s

)dt

≥ δ 3

$

h∈Z²_∗

R(h)

^−(1+δ)

(

_∞

(δσδ)^s

t

^−1/s

dt ≥ 1 3

C

₁^s

δ

^s

s 1 − s .

!

%

1

%

2

R

²

γ > 0

α ∈ R

²

$

h∈Z²_∗

1

R

l1,l2

(h)

²

% h.α %

^2−γ

< + ∞ .

δ ∈ ]0, γ/3[ R R

l1,l2

R(h)

^1+δ

% h.α % ≥ c, ∀ h ∈ Z

²_∗

,

c K

_δ⁻¹

(α) α

λ := γ − δ

$

h∈Z²∗

1

R(h)

²

% h.α %

^2−γ

≤ $

h∈Z²∗

1

R(h)

²

(cR(h)

^−(1+δ)

)

^1−γ

× 1

% h.α %

≤ B

₁

$

h∈Z²_∗

R(h)

^−(1+λ)

× 1

% h.α % . n = (n

1

, n

2

) n

i

≥ 1

$

h∈Z²_∗,|"i(h)|≤ni

1

% h.α % .

(h, h

^'

) h - = h

^'

Z

²

|% h.α % − % h

^'

.α %| ≥ min( % (h + h

^'

).α % , % (h − h

^'

).α % )

≥ c min(R(h + h

^'

)

^−(1+δ)

, R(h − h

^'

)

^−(1+δ)

)

≥ c(R

0

(2n))

^−(1+δ)

.

d := c(R

0

(2n))

^−(1+δ)

h

0 % h.α % ≥ d % x % ≤ 1/2

x % h.α %

[d, 2d[, [2d, 3d[, ..., [rd, (r + 1)d[,

r = [

_2d¹

] % h.α %

$

h∈Z²_∗,|"i(h)|≤ni

1

% h.α % ≤

$

r

k=1

1 kd ≤ 1

d (1 + ln r)

≤ 1

c R

₀

(2n)

^1+δ

(1 + ln R

₀

(2n)

^1+δ

c ) ≤ B

₂

R

₀

(n)

^1+2δ

,

B

₂

n

$

n=(n1,n2), n_i≥|"_i(h)|₀

(n

1

n

2

)

^−(2+λ)

≥ (1 + λ)

⁻²

( | %

₁

(h) |

0

| %

₂

(h) |

0

)

^1+λ

= (1 + λ)

⁻²

R(h)

^−(1+λ)

,

$

h∈Z²_∗

R(h)

^−(1+λ)

× 1

% h.α % ≤ (1 + λ)

²

$

∞ n1=1,n2=1

(n

1

n

2

)

^−(2+λ)

× $

h=(h1,h2)∈Z²_∗,|"i(h)|≤ni

1

% h.α % . B

3

, B

4

$

h∈Z²_∗

1

R(h)

²

% h.α %

^2−γ

≤ B

3

$

∞ n1=1,n2=1

(n

1

n

2

)

^−(2+λ)

× $

h=(h1,h2)∈Z²_∗,|"i(h)|≤ni

1

% h.α %

≤ B

₄

$

∞ n1=1,n2=1

(n

₁

n

₂

)

^−(2+λ)

R

₀

(n

₁

, n

₂

)

^1+2δ

= B

₄

$

∞ n1=1,n2=1

(n

₁

n

₂

)

^{−1−γ+3δ}

.

δ <

^γ₃

!

α

C

γ

(α) := $

h∈Z²_∗

1

R

l1,l2

(h)

²

% h.α %

^2−γ

.

C

0

(%

1

, %

2

) a > 0 ε > 0 a(2 − γ) = 1 − ε

(

T²

C

γ

(α)

^a

dα ≤ C

₀

εγ

^2(1+a)

( 1

γ

^a

+ 1 ε

^a

).

C

_γ

(α) ≥ K

_δ

(α)

^2−γ

δ =

^γ₂₁₋¹γ

C

₀^'

(

2

T²

C

γ

(α)

^a

dα ≥ (

T²

K

δ

(α)

^1−ε

dα ≥ C

₀^'

δ

^1−ε

1 − ε ε .

% % ²

∆

ϕ

∆

:= 1

∆

− m(∆) α = (α

1

, α

2

) ∈ T

²

% %

2

ϕ

_∆

α

S

_N^α

ϕ

_∆

(x, y) :=

N−1

$

k=0

ϕ

_∆

(x + kα

₁

, y + kα

₂

)

ϕ

∆

α

α R

^d

γ > 0 α ∈ T

²

A

γ

(α)

% S

_N^α

ϕ

∆

%

²

≤ A

γ

(α)N

^γ

.

% S

_N^α

ϕ

∆

%

²2

≤ $

h∈Z²_∗

| ϕ ˆ

∆

(h) |

²

|

N

$

−1

k=0

e

^2πik(h.α)

|

²

≤ $

h∈Z²_∗

| ϕ ˆ

∆

(h) |

²

inf(N

²

, 1

|| h.α ||

²

)

≤ N

^2γ

$

h∈Z²_∗

| ϕ ˆ

∆

(h) |

²

1

|| h.α ||

^2(1−γ)

. γ > 0

α !

{ α : A

γ

(α) < + ∞} ϕ

∆

A

γ

= C

1

γ2

A

γ

(ϕ

∆i

, i = 1, · · · , d) ϕ

∆i

α = (α

1

, α

2

) R

^d

T

²

(x, y ) → (x + α

1

, y + α

2

) Φ = (ϕ

∆_i

)

i=1,···,d

1

_[0,¹

2[

−

¹₂

ϕ

∆

Φ(x, y) = (ϕ

1

(x), ϕ

2

(y)) ϕ

1

ϕ

2

(α

1

, α

2

)

α

1

f

T

¹

ε > 0 x !

n−1

k=0

f (x + kα

1

) = O(n

^ε

) Φ(x, y) = (2.1

_[0,¹

2[

(x) − 1), 2.1

_[0,¹

2[

(y) − 1) Z

²

Z

²

(α

1

, α

2

)

T

²

Z

²

τ

Φ

: (x, y, n) → (x + α

1

, y + α

2

, n + Φ(x, y)).

(Φ, τ ) Φ

σ

α = (α

₁

, α

₂

) Φ

Z

²

Φ

α (k

n

)

lim sup

n

% S

_k^α_n

Φ %

²

√ n > 0.

Φ : T

²

→ R

²

Φ(x) = Φ(x

1

, x

2

) = (ϕ(x

1

), ϕ(x

2

)), where ϕ := 1

_[0,¹

5]

− 1 5 . T

²

x → x + α α = (α

1

, α

2

)

(x

1

, x

2

) T

²

|

N−1

$

k=0

ϕ(x

1

+ kα

1

) | + |

N−1

$

k=0

ϕ(x

2

+ kα

2

) |

^N

−→

^→+∞

+ ∞ . Φ

Φ = (ϕ

₁

, ϕ

₂

) = (1

_R1

− 1

5 , 1

_R2

− 1 5 ), R

1

= [0,

¹₅

] × [0, 1] R

2

= [0, 1] × [0,

¹₅

]

α = (α

₁

, α

₂

) R

²

Z α + Z

²

R

²

(x

1

, x

2

) ∈ T

²

| S

_N^α1

ϕ

1

(x

1

) | | S

_N^α2

ϕ

2

(x

2

) | N

(k(n))

n≥1

k(1) = 1

k(n + 1) ≥ 10 × ln 3

ln 2 × k(n), ∀ n ≥ 1.

α α = (

$

∞ n=1

3

^−k(2n+1)

,

$

∞ n=1

2

^−k(2n)

).

p

n

= 2 n p

n

= 3 n (β

n

)

n≥0

α

β

n

:= ( $

m<ⁿ₂

3

^−k(2m+1)

, $

m≤ⁿ₂

2

^−k(2m)

).

n ≥ 1 β

n

Q

n

= p

^k(n_n−1⁻¹⁾

p

^k(n)n

α = β

n

+ ε

n

ε

n

= ( !

m≥ⁿ₂

3

^−k(2m+1)

, !

m>ⁿ₂

2

^−k(2m)

) k(%) % > n

| ε

n

| ≤ $

m≥ⁿ₂

3

^−k(2m+1)

+ $

m>ⁿ₂

2

^−k(2m)

≤ 4 × 2

^−k(n+1)

.

β

n

(

^p_qnⁿ

,

^p_q^#n_#

n

) q

n

= 3

^k(n−1)

q

_n^'

= 2

^k(n)

n q

n

= 3

^k(n)

q

^'_n

= 2

^k(n−1)

n Q

_n

= q

_n

q

^'_n

r(j) r

^'

(j ) j q

_n

q

_n^'

j /→ (r(j ), r

^'

(j)) [0, q

n

q

_n^'

− 1] [0, q

n

− 1] × [0, q

_n^'

− 1]

{ jβ

n

mod Z

²

, j ∈ Z} {

_q^j_n

mod 1, 0 ≤ j ≤ q − 1 } × {

_q^j#n^#

mod 1, 0 ≤ j

^'

≤ q

_n^'

− 1 }

q ≤ Q

n

d(qα, qβ

n

) = q | ε

n

| ≤ Q

n

| ε

n

| = p

^k(n_n−1⁻¹⁾

p

^k(n)_n

× 4 × 2

^−k(n+1)

→ 0,

Z α + Z

²

R

²

p ≥ 1

p p = 2n + 1 s

j,2n+1

j = 0, ..., 2

^k(2n)

− 1

jβ

2n+1

+ [0, 1] × { 0 } p = 2n s

j,2n

j = 0, ..., 3

^k(2n−1)

− 1 jβ

2n

+ { 0 } × [0, 1]

q q = ap

^k(n_n−1⁻¹⁾

+ j a j 0 ≤ j <

p

^k(n−1)_n−1

qβ

n

s

j,n

q ≤ Q

²_n

qα s

j,n

d

_T²

(qα, s

j,n

) = d

_T²

(q(β

n

+ ε

n

), s

j,n

) ≤ 0 + q | ε

n

| ≤ Q

²_n

| ε

n

| .

δ

n

:= Q

²_n

| ε

n

|

B

j,n

≤ 2δ

n

s

j,n

r

_n

s

_j,n

s

i,n

p

⁻_n−1^k(n−1)

4 × δ

n

≤ p

^2k(n_n−1⁻¹⁾

p

^2k(n)_n

× 16 × 2

^−k(n+1)

≤ 3

^4k(n)+3

2

^−k(n+1)

≤ p

^−6k(n)+3_n−1

< 1

2 p

^−k(n−1)_n−1

, B

j,n

j = 0, ..., p

^k(n−1)n

− 1

R

ⁱ

s

j,n

s

j,n

R

ⁱ

− x J

x

= J

n

J

n

+ 1 J

n

= )

1/5 rn

*

t

n

=

¹₅

− J

n

r

n

ρ

n

= min(t

n

, r

n

− t

n

) ρ

_n

= min

j∈Z

&

&

&

&

&

1

5 − j p

^k(n_n−1⁻¹⁾

&

&

&

&

& ≥ 1 5 × p

^k(n_n−1⁻¹⁾

.

I

_n

=

p^k(n−1)_n−1 −1

'

j=0

B

_j,n

, K

_n

= I

_n

± ρ

_n

e

_n

, E

_n

= I

_n

∪ K

_n

,

e

n

= (1, 0) n (0, 1) n E

n

3p

^k(n_n−1⁻¹⁾

× ( B

j,n

) m

n

= 12 × p

^k(n_n−1⁻¹⁾

δ

n

N [Q

²_n−1

, Q

²_n

[

n n x ∈ T

²

\ E

_n

M

x,N

q < N x + qα ∈ R

²

s

j,n

d

_T²

(x + s

j,n

, { 0 } × [0, 1]) = d

_T²

(x + jβ

n

, { 0 } × [0, 1])

= d

_T²

(x, − jβ

n

+ { 0 } × [0, 1]) = d

_T²

(x, s

i,n

)

i { 0, ..., p

^k(n_n−1⁻¹⁾

− 1 } x I

n

d

_T²

(x, s

i,n

) > 2δ

n

x + s

j,n

{ 0 } × [0, 1]

2δ

n

d

_T²

(x + s

j,n

, { 1

5 } × [0, 1]) = d

_T²

(x + jβ

n

, { 1

5 } × [0, 1])

= d

_T²

(x, − jβ

n

+ { 1

5 } × [0, 1])

= d

_T²

(x, − jβ

n

+ { J

n

r

n

+ t

n

} × [0, 1])

= d

_T²

(x, ( − jβ

n

+ (J

n

p

^k(n_n−1⁻¹⁾

, 0)) + { t

n

} × [0, 1])

= d

_T²

(x, s

i,n

+ t

n

e

n

) x K

_n

d

_T²

(x, s

_i,n

+ t

_n

e

_n

) > 2δ

_n

x + s

j,n

{

¹₅

} × [0, 1]

2δ

n

x + B

j,n

R

²

R

2

j < p

^k(n_n−1⁻¹⁾

q < N qα

B

j,n

A A + 1 A := [(N − 1) × p

⁻_n−1^k(n−1)

]

J

x

A ≤ M

x,N

≤ J

x

(A + 1).

| S

N

ϕ

2

(x) | = & & M

x,N

−

¹₅

N & & J

x

= J

n

1

5 N − M

_x,N

≥ 1

5 N − J

_x

(A + 1) ≥ 1

5 N − J

_n

( N

p

^k(n−1)_n−1

+ 1)

= ( 1

5 − J

_n

p

^k(n_n−1⁻¹⁾

)N − J

n

≥ ρ

n

N − J

n

≥ 1

5 × p

^k(n−1)_n−1

Q

²_n−1

− 1/5

p

^−k(n−1)_n−1

= p

^k(n−1)_n−1

5 (p

^2k(n−2)_n−2

− 1) J

x

= J

n

+ 1

M

x,N

− 1

5 N ≥ (J

n

+ 1)A − 1

5 N ≥ p

^k(n_n−1⁻¹⁾

5 (p

^2k(n−2)_n−2

− 1) − 1.

(x -∈ E

n

) = ⇒ | S

N

ϕ

2

(x) | ≥ p

^k(n−1)_n−1

5 (p

^2k(n)_n

− 1) − 1.

n S

N

ϕ

1

(x)

$

n≥1

m(E

n

) ≤ 12 $

n≥1

p

^k(n_n−1⁻¹⁾

δ

n

= 12 $

n≥1

p

^k(n_n−1⁻¹⁾

Q

²_n

| ε

n

|

≤ 12 $

n≥1

p

^k(n−1)_n−1

p

^2k(n−1)_n−1

p

^2k(n)_n

× 4 × 2

^−k(n+1)

< + ∞ ,

x T

²

E

n

N→+∞

lim | S

N

ϕ(x) | = + ∞ .

!

α

ε > 0

N

_ε

N ≥ N

_ε

min {% qα % : 0 < q ≤ N } ≤ εN

⁻¹²

. N ∈ [Q

n

, Q

n+1

[

min {% qα % : 0 < q ≤ N } ≤ | ε

n−1

|

| ε

n

| N

¹²

≤ ( $

m≥ⁿ₂

3

^−k(2m+1)

+ $

m>ⁿ₂

2

^−k(2m)

)Q

_n+1¹²

.

n

( $

m≥ⁿ₂

3

^−k(2m+1)

+ $

m>ⁿ₂

2

^−k(2m)

)Q

_n+1¹²

≤ 2 $

m≥ⁿ₂

3

^−k(2m+1)

Q

_n+1¹²

≤ 4 × 3

^−k(n+1)

× 2

^12k(n)

3

^12k(n+1)

n

( $

m≥ⁿ₂

3

^−k(2m+1)

+ $

m>ⁿ₂

2

^−k(2m)

)Q

_n+1¹²

≤ 2 $

m>ⁿ₂

2

^−k(2m)

Q

_n+1¹²

≤ 4 × 2

^−k(n+1)

3

^12k(n)

2

^21k(n+1)

.

| ε

n

| × Q

_n+1¹²

0 n → ∞ T

²

τ

ϕ

: T

²

× R

²

→ T

²

× R

²

ϕ α

Z

^d

d

2