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Parallelepipeds, Nilpotent Groups, and Gowers Norms

a r X i v :m a t h /0606004v 1 [m a t h .C O ] 31 M a y 2006

Z /N Z

(k ?1)

k ≥2

{(x 00,x 01,x 10,x 11)∈(Z /N Z )4:x 00?x 01?x 10+x 11=0}

(Z/N Z)4f:Z/N Z→C f U2

f(x00)f(x10)f(x11).

f 4U

(x00,x01,x10,x11)∈P

U k k≥3

{0,1}k

?=?1...?k∈{0,1}k

|?|=?1+···+?k.

x=(x1,...,x k)∈(Z/N Z)k?∈{0,1}k

?·x=?1x1+···+?k x k.

C:C→C n∈N∪{0}ξ∈C

C nξ= ξn

f(x+m)

f(x+p)f(x+m+p)f(x+n+p)

· U

U k

X4X8

X X[2]=X×X×X×X X[3]

X[2]x=(x0,x1,x2,x3)x=(x00,x01,x10,x11) X[3]x=(x000,x001, (x111)

x∈X[2]x=(x i:0≤i≤3)x=(x?:?∈{0,1}2)

X[3]

{0,1}2

x X[2]

X[2]

x→(x10,x11,x00,x01)x→(x10,x00,x11,x01)

X[3]

π:X→Yπ[2]:X[2]→Y[2]

π[2](x)=π(x00)π(x01)π(x10)π(x11).

π[3]:X[3]→Y[3]

X X P X[2]

～X2(x00,x01)～(x10,x11) (x00,x01,x10,x11)∈P

(x00,x01,x10,x11)∈P(x00,x10,x01,x11)∈P

x00,x01,x10∈X

x11∈X(x00,x01,x10,x11)∈P

P x11

X/～P

B(x,y)X2 x,y

～

x0,x1∈X(x0,x0,x1,x1)∈P

P X[2]

～x00,x01,x10,x11∈X(x00,x01,x10,x11)∈P x00,x01 = x10,x11 x00,x10 = x01,x11

(x,x)x∈X～

～x0,x1∈X(x0,x1,x0,x1)∈P

(x0,x0,x1,x1)∈P

～

(x,x),x∈B1B X · U2

X F(X)

X P

X f∈F(X)X

x∈P f(x00)f(x10)f(x11)≥0.

f P:= x∈P f(x00)f(x10)f(x11) 1/4,

f→ f P F(X)

F,G X2

x∈P F(x00,x01)G(x10,x11)= z∈B (x,y)∈X2

F(x,y) (x,y)∈X2 x,y =z G(x,y) .

x,y =z

F(x10,x11)≥0.

F,G∈F(X2)

x∈P F(x00,x01)G(x10,x11)

≤ x∈P F(x00,x01)G(x10,x11) 1/2 F(x00,x01)=f(x00)

f01(x01)

f01(x01)G(x10,x11)=f10(x10)

f01(x01)f11(x01)

f00(x01)

n M0≤λi≤n i=1,...,n2 n2

i=1λi=Trace(M)=n2n2 i=1λ2i=Trace(M2)=n3.

λi0n M2=n·M

P G:= g=(g00,g01,g10,g11)∈G[2]:g00g?101g?110g11=1

= (g,gs,gt,gst):g,s,t∈G G

X G P G

Gπ:X→G P

P Gπ[2]:X[2]→G[2]

P={x∈X[2]:π[2]x∈P G}

= x∈X[2]:π(x00)π(x01)?1π(x10)?1π(x11)=1 .

P Xπ

a,b · b,c = a,c a,b,c∈X.

s,t∈B a∈X

b∈X a,b =s c∈X b,c =t

a,c a,b,c s t ,a′,b′,c′∈X a′,b′ =s c′,d′ =t

a,a′ = b,b′ = c,c′ a,c = a′,c′

a,b b,a

s,t,a,b,c

d∈X a,d =t(a,b,d,c)∈P

d,c = a,b =s t·s= a,d · d,c = a,c =s·t

i∈Xπ:X→B

π(x)= i,x x∈X.

X P

B i∈Xπ:X→B

x∈X[2]

π(x00)·π(x01)?1·π(x10)?1·π(x11)= i,x00 · i,x01 ?1· i,x10 ?1· i,x11

= x00,x01 · x10,x11 ?1

1B x∈P P

P Bπ[2]

PπX

πP P X x,y∈X

(x,x,x,y)∈P

π(x)=π(y)

x,y =1B

a,b,c∈X(a,b,c,x)∈P(a,b,c,y)∈P

≡≡

B π:X→B X/≡

· P

a,b∈Xπ

a b P f=1{a}?1{b}

818?1

f P=0

G G[2,2]G[2]

G[2,2]:= (g,g,g,g):g∈G .

G[2,1]G[2]

(g,g,1,1);(g,1,g,1);(g,g,g,g)g∈G.

G[2,1]G

{0,1}2

G[2]g

{0,1}21

G G2

G2G[g,h]g,h∈G [g,h]=ghg?1h?1G3G

G[g,u]g∈G u∈G2

G[2,1]= g∈G[2]:g00g?101g?110g11∈G2 = (g,gh,gk,ghku):g,h,k∈G,u∈G2 .

G G[2,1]P G

G[2,1]:= g∈G[2]:g00g?101g?110g11=1 ={(g,gs,gt,gst):g,s,t∈G}.

G F G G2F

G B=G/Fπ:G→B

P G

P= g∈G[2]:π[2](g)∈(G/F)[2,1]={g∈G[2]:g00g?101g?110g11∈F}.

P G[2]P=G[2,1]F[2]

X[3]{0,1}3

{0,1}3

xσ{0,1}2x

x=(x′,x′′)X[3]

X[2]×X[2]

P Q X[3]

x∈Qσ{0,1}3xσ∈P

Q{0,1}3

≈≈

P x′≈x′′(x′,x′′)∈Q

x000,x001,x010,x011,x100,x101,x110

(x000,x001,x010,x011),(x000,x010,x100,x110),(x000,x001,x100,x101)∈P,

x111∈X

(x000,x001,x010,x011,x100,x101,x110,x111)∈Q.

Q x111

X P Q (P,Q)

P/≈P

x∈P[x][x]Q

x000, (x110)

x111∈X(x010,x110,x011,x111)∈P

∈X[3]P

x′111∈X(x000,x001,x010,x011,x100,x101,x110,x′111)∈Q

≈≈

X(P,Q)

x,y∈X[x,y,x,y] x,y

(a,a,a,a)a∈X

≈

x,y = x′,y′ x=(x,y,x′,y′)∈P

≈(x,x)∈Q

(x,y,x,y,x′,y′,x′,y′)∈Q[x,y,x,y]=[x′,y′,x′,y′]

X P Q X[3]

x,x′∈X

(x,x,x,x′)∈P[x,x,x,x′]=1P

a,b,c∈X(a,b,c,x)∈P(a,b,c,x′)∈P [a,b,c,x]=[a,b,c,x′]

a,b,c∈X(a,b,c,x)∈P(a,b,c,x′)∈P[a,b,c,x]= [a,b,c,x′]

x,x′ =1B(x,x,x,x′)∈P[x,x,x,x′]=1P a,b,c∈X(a,b,c,x)∈

P[c,c,c,c]=1P=[x,x,x,x′][c,x,c,x]= [c,x,c,x′][a,b,a,b]=[c,x,c,x]=[c,x,c,x′] [a,b,c,x]=[a,b,c,x′]x,x′

a,b,c∈X(a,b,c,x)∈P[a,b,c,x]= [a,b,c,x′][x,x,x,x′]=[x,x,x,x]=1P

x,x′∈X x≡

x′x,x′

≡

x′∈X[3]x

′∈Q(P,Q)

≡

r:X→Y P Y Y X/≡

Q Y P Q r[2]r[3]P Q

P Y Q Y

Q Q Y

(P Y,Q Y)Y

≡

Q Y

(P,Q)X

f∈F(X)

1/8.

∈Q ?∈{0,1}3C|?|f(x?)

f→ f Q F(X)

(P,Q)

≈

X P X Q?X[3]

≈

F X4

x f(x′′)≥0.

Gα{0,1}3g∈G g[3,α]G[3]

g[3,α] ?= g?∈α

g[3,η]g∈Gη∈{0,1}3G[3]α={0,1}3 g[3,α](g,g,...,g)

G[3,3]G[3]

(g,g,...,g)g∈G

G[3,1]G[3]

g[3,e]g∈G e{0,1}3

G[3,2]G[3]

g[3,f]g∈G f{0,1}3

G[3,1]= g

∈G[3]:g

G F G

P=G[2,1]F[2]Q=G[3,2]F[3,1].

(P,Q)X

Gα,β{0,1}3 g,h∈G g[3,α]h[3,β]G[3]

g[3,α],h[3,β] =[g,h][3,α∩β].

G[3,2]?G[3,1]

2?G[3]3

g∈Gη∈{0,1}3g[3,η]∈G[3,2]g∈G3

G[2,2]G[2,1]

2G[2]3G[2,1]

G[3]=G[2]×G[2]

G[3,2]= g

=(g′,g′′)∈G[3,2] g′g′′G[2,1]g∈G[2,1](g,g)∈G[3,2]

G[3,2]= (g′,g′′)∈G[2,1]×G[2,1]:g′g′′?1∈L ,

L= g∈G[2,1]:(g,1)∈G[3,2]}.

L=K K?L

G[3,2]G[3]

G[3]G[3,2]

K?L

g∈G3η∈{0,1}3g[3,η]∈G[3,2]

g∈G g[3,η]∈G[3,2]η

g∈G3G[3,2]η=111

(1,1,1,g)G[2,2]G[2,1]

G[2]3

(1,1,1,g)=(h,h,h,h).u.v h∈G,u∈G[2,1]

v∈G[3]3.

G2h∈G2

(h,h,h,h)∈G[2,1]

2(h,h,h,h).u u

h=1

G3G2G2/G3ˉu

(G2/G3)[2]u G3ˉu(G2/G3)[2,1]1

1u11∈G3 g∈G3

G P=G[2,1]Q=G[3,2](P,Q)

G G2

G F G

G2?F?Z(G).

P=G[2,1]F[2]Q=G[3,2]F[3,1](P,Q)

Q Q G[3]

G[2,2]F[2,1]

G[2,1]F[2]

Q={(g′,g′′)∈G[2]×G[2]:g′∈G[2,1]F[2],g′g′′?1∈G[2,2]F[2,1]}.

P P≈

Q′G[3]g

P Q′F[3]Q′/F[3]

(G/F)[3,2]G/F

Q′=G[3,2]F[3]

g=j j F[3]F

g h

(1,1,1,g)∈G[2,2]F[2,1]F[2,1]

g=1

(P,Q)

F=G2

(P,Q)P Q

G[2]2?G[2,1]

G[2,1]G[2]2=G[2,1]=P G[3,2]G[3,1]

2=G[3,2]=

G/G32

P G[3]3Q G[3]3P

(G/G3)[2,1]G[2]Q(G/G3)[3,2]G[3]

G F G

G2?F?Z(G)Γ

G X=G/Γ

G/(Γ∩F)G F/(Γ∩F)FΓ/(Γ∩F)Γ

Γ∩F={1}

G F G

G2?F?Z(G)

ΓG

Γ∩F={1}.

P=G[2,1]F[2]Q=G[3,2]F[3,1]π:G→G/Γ

P X=π[2](P)Q X=π[3](Q).

(P X,Q X)X

G,F,Γ,X,P X,Q X

g=(g00,g01,g10,g11)∈G[2]π[2](g)∈P X g′11∈Gπ(g′11)=π(g11) (g00,g01,g10,g′11)∈P

h∈Pπ[2](h)=π[2](g)

γ∈Γ[2]g=hγΓ

θ∈Γ(γ00,γ01,γ10,γ11θ)∈Γ[2,1]g′11=g11θ

P X

X B=G/FΓ

x P X Q

x,y,z P X(x,y)(y,z)

Q X g=(h′,h′′) Q

π[2](g′)=x;π[2](g′′)=π[2](h′)=yπ[2](h′′)=z.

γ=g′′?1h′γ∈Γ[2]∩G[2,1]F[2]γ00γ?101γ?110γ11∈Γ∩F={1}γ∈Γ[2,1](g′γ,h′)∈Q (g′γ,h′′)∈Q X (x,z)(x,z)∈Q≈Q X P X

∈G[3]π[3](g g

G Q

G g111

G X

Q X x

)=x)=y=g u

?= g·x??∈α

(P,Q)

X(P,Q)G G Q

x→g·x X x∈P g[2]·x

g[2]·x≈x

(P,Q)

X G x→g·x X

x∈Q φ{?∈{0,1}3:?3=1}

g∈G x=(x′,x′′) x′,x′′∈P g[3,φ]·x

∈Q

x→g·x X x

∈Q x∈P x

∈Q g[2]·x∈P g[2]·x≈x

(P,Q)

X G2

G[3]X[3]g∈G x

=(x′,x′′)x,x′∈P g[2]·x′x′

(g[2]·x′,x′)∈Q(g[2]·x′,x′′)∈Q Q f {0,1}3g[3,f]·x∈Q Q G[3,2] g∈G3(1,...,1,g)∈G[3,2]x∈X

(x,...,x,x)∈Q(x,...,x,g·x)∈Q Q

g·x=x g=1G2

G G(X)

G(X)G

φ:B→F

h X

h·x=φ(π(x))·x.

h G(X)G

(P,Q)X

Pπ:X→B

x,y∈X x,y

(x,y)～

x,y =π(y)π(x)?1.

b∈B F b b F b:=π?1({b})

≈x[x]

P/≈P

P s s∈B

X s:= (x01,x10)∈X2: x01,x02 =s}?X2.

P s:= x∈P: x00,x01 =s ?X[2].

Q s:= y

∈X[2]s X s X

X x

u,v P s(x0,x1,x2,x3)

u x2,x3 =s x4x5X(x2,x3,x4,x5)

v uv(x0,x1,x4,x5)

s∈B x y X

P s≈ x00,x10 = y00,y10

q s:P s→B

q s([x])= x00,x10 x∈P s.

P s B

x∈P x00,x01 =sπ(x10)=π(x00)

(P,Q)X

x∈P[x00,x00,x01,x01]=[x10,x10,x11,x11]

[x00,x01,x00,x01]=[x10,x11,x10,x11]

x y[x,x,y,y]=[x,y,x,y]=1

(x,x)∈Q

Q(x,x,x,y)∈P x00,x01,x10,x11(x00,x01,x10,x11)∈P

x00,x01,x10,x11

[x00,x01,x10,x11]=[x00,x10,x01,x11].

y∈X[x00,x01,x10,x11]=[x00,x01,x01,y] (x00,x01,x10,x11,x00,x01,x01,y)∈Q(x00,x10,x01,x11,x00, x01,x01,y)∈Q[x00,x10,x01,x11]=[x00,x01,x01,y]=[x00,x01,x10,x11]

F X

B1F

q1:P1→B

F≈

x0,x1,x2,x3,x4,x5

[x0,x1,x2,x3][x2,x3,x4,x5]=[x0,x1,x4,x5].

F X

x∈X u∈F u

y∈X [x,x,x,y]=u y

x∈X u∈F u·x X

[x,x,x,y]=u

(u,x)→u·x F X

1P·x=x x∈X u,v∈F x∈X(vu)·x=v·(u·x)y=u·x z=v·y=v·(u·x)

v=[u·x,u·x,u·x,v·(u·x)]=[x,u·x,x,u·x][x,u·x,x,v·(u·x)]=[x,u·x,x,v·(u·x)] [x,x,x,v·(u·x)]=[x,x,x,u·x][x,u·x,x,v·(u·x)]=uv=vu F

F x,y

u∈F u·x=y u=[x,x,x,y]

(x,u·x,v·x,w·x)

F wu?1v?1F

x∈P u∈F(u·x00,u·x01,x10,x11)∈P[u·x00,u·x01,x10,x11]=[x]

F[2,1]

w=[x,x,x,w·x]=[x,x,x,u·x][x,u·x,v·x,w·x][v·x,w·x,x,w·x]

=u[x,u·x,v·x,w·x][v·x,x,w·x,w·x]

=u[x,u·x,v·x,w·x][v·x,x,x,x][x,x,w·x,w·x]

=u[x,u·x,v·x,w·x][v·x,x,x,x]=u[x,u·x,v·x,w·x][x,x,x,v·x]

=u[x,u·x,v·x,w·x]v.

x∈P u∈F

[x00,x01,x00,x01]=[x10,x11,x10,x11][x00,x00,u·x00,x00]= u?1=[x01,x01,u·x01,x01][x00,x01,u·x00,u·x01]=[x00,x01,x00,x01]= [x10,x11,x10,x11]

F[2]X[2]

Q F[3,1] F[3]

x∈P u∈F[2]x u·x u∈F[2,1]

u′11(u00,u01,u10,u′11)∈F[2,1][u00·x00,u01·x01,u10·x10,u′11·x11]=[x]=[u00·x00,u01·x01,u10·x10,u11·x11]

u′11·x11=u11·x11F

u′11=u11

Q s X s

X s

(x,y)(x′,y′)X s

u∈F x′=u·x y′=u·y

F G

u∈F u[2]∈F[2,1]F?G

g∈G u∈F f e

f∩e={111}g[3,f]u[3,e]Q

[g;u][3,e]x∈Q

(x000,x001,...,x110,[g;u]·x111)∈Q[g;u]·x111=111 [g;u]

i X

π:X→Bπ(x)= i,x x∈X

G G F G

XΓi∈X G X

G/Γ(P,Q)X

(P X,Q X)G,FΓ

(P,Q)

G X

G,FΓ

F G F

XΓ∩F={1}

F G2G

G[3,2]?G[3,1]

2G[3,1]

G[2,1]

B g p

g∈G2x,y∈X u,v∈F g·x=u·x g·y=v·y

(x,y,x,y)∈P(g,g,1,τ)∈G[2,1]

2[x,y,x,y]=[g·x,g·y,x,y]=

[u·x,v·y,x,y]u=v