Multiple correlation sequences and applications
Nikos Frantzikinakis
University of Crete, Greece
Bedlewo, November 2015
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
1 / 17
Three interconnected topics
1
Understand the structure of multiple correlation sequences
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt.
2
Convergence criterions for weighted ergodic averages
N
1X
wn · T1n f1 · . . . · T`n f` .
N
n=1
3
Multiple recurrence and convergence results for (shifts) of the sets
Sa,b = {n ∈ N : number of distinct prime factors ≡ a
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
mod b}.
Bedlewo, November 2015
2 / 17
Three interconnected topics
1
Understand the structure of multiple correlation sequences
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt.
2
Convergence criterions for weighted ergodic averages
N
1X
wn · T1n f1 · . . . · T`n f` .
N
n=1
3
Multiple recurrence and convergence results for (shifts) of the sets
Sa,b = {n ∈ N : number of distinct prime factors ≡ a
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
mod b}.
Bedlewo, November 2015
2 / 17
Three interconnected topics
1
Understand the structure of multiple correlation sequences
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt.
2
Convergence criterions for weighted ergodic averages
N
1X
wn · T1n f1 · . . . · T`n f` .
N
n=1
3
Multiple recurrence and convergence results for (shifts) of the sets
Sa,b = {n ∈ N : number of distinct prime factors ≡ a
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
mod b}.
Bedlewo, November 2015
2 / 17
Background (Multiple recurrence)
Theorem (Furstenberg, Katznelson 1979)
Let T1 , . . . , T` be commuting mpt acting on the probability space
(X , X , µ). Then for every A ∈ X with µ(A) > 0 there exists n ∈ N s.t.
µ(A ∩ T1−n A ∩ · · · ∩ T`−n A) > 0.
Theorem (Multidimensional Szemerédi)
Let E ⊂ Zd with d(E) > 0. Then for every v1 , . . . , v` ∈ Zd we have
v , v + nv1 , . . . , v + nv` ∈ E
for some v ∈ Zd and n ∈ N.
Bergelson, Leibman (1996), Leibman (1998): Extension to
polynomial iterates and tranformations generating a nilpotent group.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
3 / 17
Background (Multiple recurrence)
Theorem (Furstenberg, Katznelson 1979)
Let T1 , . . . , T` be commuting mpt acting on the probability space
(X , X , µ). Then for every A ∈ X with µ(A) > 0 there exists n ∈ N s.t.
µ(A ∩ T1−n A ∩ · · · ∩ T`−n A) > 0.
Theorem (Multidimensional Szemerédi)
Let E ⊂ Zd with d(E) > 0. Then for every v1 , . . . , v` ∈ Zd we have
v , v + nv1 , . . . , v + nv` ∈ E
for some v ∈ Zd and n ∈ N.
Bergelson, Leibman (1996), Leibman (1998): Extension to
polynomial iterates and tranformations generating a nilpotent group.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
3 / 17
Background (Mean convergence)
After a long series of partial results by Furstenberg, Conze-Lesigne,
Bergelson, Leibman, Furstenberg-Weiss, Rudolph, Zhang, Host-Kra,
Ziegler, F.-Kra ...
Theorem (Tao 2008)
Let T1 , . . . , T` be commuting mpt acting on the probability space
(X , X , µ). Then for every f1 , . . . , f` ∈ L∞ (µ) the averages
N
1X n
T1 f1 · · · T`n f`
N
n=1
converge in L2 (µ).
Alternate proofs given by [Towsner 09], [Austin 10], [Host 09].
Walsh (2012): Extension to polynomial iterates and transformations
generating a nilpotent group.
But these methods do not not work for weighted averages.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
4 / 17
Background (Mean convergence)
After a long series of partial results by Furstenberg, Conze-Lesigne,
Bergelson, Leibman, Furstenberg-Weiss, Rudolph, Zhang, Host-Kra,
Ziegler, F.-Kra ...
Theorem (Tao 2008)
Let T1 , . . . , T` be commuting mpt acting on the probability space
(X , X , µ). Then for every f1 , . . . , f` ∈ L∞ (µ) the averages
N
1X n
T1 f1 · · · T`n f`
N
n=1
converge in L2 (µ).
Alternate proofs given by [Towsner 09], [Austin 10], [Host 09].
Walsh (2012): Extension to polynomial iterates and transformations
generating a nilpotent group.
But these methods do not not work for weighted averages.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
4 / 17
Multiple correlation sequences
Such problems lead to the study of the multiple correlation sequences:
Definition
A multiple correlation sequence of order ` is any sequence of the form
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt and f0 , . . . , f` ∈ L∞ (µ).
Main problem (vaguely stated):
Problem
Determine the structure of the multiple correlation sequences (C` (n)).
If we allow errors small in density: Solved for all ` ∈ N.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
5 / 17
Multiple correlation sequences
Such problems lead to the study of the multiple correlation sequences:
Definition
A multiple correlation sequence of order ` is any sequence of the form
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt and f0 , . . . , f` ∈ L∞ (µ).
Main problem (vaguely stated):
Problem
Determine the structure of the multiple correlation sequences (C` (n)).
If we allow errors small in density: Solved for all ` ∈ N.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
5 / 17
Multiple correlation sequences
Such problems lead to the study of the multiple correlation sequences:
Definition
A multiple correlation sequence of order ` is any sequence of the form
Z
C` (n) := f0 · T1n f1 · . . . · T`n f` dµ, n ∈ N,
where T1 , . . . , T` : X → X are commuting mpt and f0 , . . . , f` ∈ L∞ (µ).
Main problem (vaguely stated):
Problem
Determine the structure of the multiple correlation sequences (C` (n)).
If we allow errors small in density: Solved for all ` ∈ N.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
5 / 17
Single correlations (C1 (n) =
R
f · T n g dµ)
Examples
(e2πinα )n∈N , (
P∞
k =1 ck e
2πinαk )
n∈N
where
P∞
k =1 |ck |
< ∞, αk ∈ R.
Theorem (Herglotz 1911)
There exists a complex measure σ = σf ,g,T with bounded variation s.t.
Z
Z
f · T n g dµ = e2πint dσ(t), n ∈ N.
Corollary (Structure of single correlation sequences)
For a given system (X , µ, T ) and f , g ∈ L∞ (µ) we have
Z
f · T n g dµ = F (an ) + E(n), n ∈ N,
where F ∈ C(T∞ ), a ∈ T∞ , and limN−M→∞
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
1
N−M
PN−1
n=M
|E(n)| = 0.
Bedlewo, November 2015
6 / 17
Single correlations (C1 (n) =
R
f · T n g dµ)
Examples
(e2πinα )n∈N , (
P∞
k =1 ck e
2πinαk )
n∈N
where
P∞
k =1 |ck |
< ∞, αk ∈ R.
Theorem (Herglotz 1911)
There exists a complex measure σ = σf ,g,T with bounded variation s.t.
Z
Z
f · T n g dµ = e2πint dσ(t), n ∈ N.
Corollary (Structure of single correlation sequences)
For a given system (X , µ, T ) and f , g ∈ L∞ (µ) we have
Z
f · T n g dµ = F (an ) + E(n), n ∈ N,
where F ∈ C(T∞ ), a ∈ T∞ , and limN−M→∞
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
1
N−M
PN−1
n=M
|E(n)| = 0.
Bedlewo, November 2015
6 / 17
Single correlations (C1 (n) =
R
f · T n g dµ)
Examples
(e2πinα )n∈N , (
P∞
k =1 ck e
2πinαk )
n∈N
where
P∞
k =1 |ck |
< ∞, αk ∈ R.
Theorem (Herglotz 1911)
There exists a complex measure σ = σf ,g,T with bounded variation s.t.
Z
Z
f · T n g dµ = e2πint dσ(t), n ∈ N.
Corollary (Structure of single correlation sequences)
For a given system (X , µ, T ) and f , g ∈ L∞ (µ) we have
Z
f · T n g dµ = F (an ) + E(n), n ∈ N,
where F ∈ C(T∞ ), a ∈ T∞ , and limN−M→∞
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
1
N−M
PN−1
n=M
|E(n)| = 0.
Bedlewo, November 2015
6 / 17
Single correlations (C1 (n) =
R
f · T n g dµ)
Examples
(e2πinα )n∈N , (
P∞
k =1 ck e
2πinαk )
n∈N
where
P∞
k =1 |ck |
< ∞, αk ∈ R.
Theorem (Herglotz 1911)
There exists a complex measure σ = σf ,g,T with bounded variation s.t.
Z
Z
f · T n g dµ = e2πint dσ(t), n ∈ N.
Corollary (Structure of single correlation sequences)
For a given system (X , µ, T ) and f , g ∈ L∞ (µ) we have
Z
f · T n g dµ = F (an ) + E(n), n ∈ N,
where F ∈ C(T∞ ), a ∈ T∞ , and limN−M→∞
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
1
N−M
PN−1
n=M
|E(n)| = 0.
Bedlewo, November 2015
6 / 17
R
Double correlations ( f · T n g · T 2n h dµ)-Nilsequences.
Examples
2
(e2πi(n α+nβ) )n∈N , (e2πi[nα]nβ · φ(nα))n∈N for some φ ∈ C(T).
Also finite products of such sequences.
All these are particular cases of:
Definition (`-step nilsequences)
X = G/Γ where G is a `-step nilpotent group and Γ is a discrete
cocompact subgroup;
F ∈ C(X ) (or Riemann integrable), a ∈ G;
(F (an Γ))n∈N is called a basic `-step nilsequence;
`-step nilsequence= a uniform limit of basic `-step nilsequences.
1-step nilsequence = almost periodic sequence.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
7 / 17
R
Double correlations ( f · T n g · T 2n h dµ)-Nilsequences.
Examples
2
(e2πi(n α+nβ) )n∈N , (e2πi[nα]nβ · φ(nα))n∈N for some φ ∈ C(T).
Also finite products of such sequences.
All these are particular cases of:
Definition (`-step nilsequences)
X = G/Γ where G is a `-step nilpotent group and Γ is a discrete
cocompact subgroup;
F ∈ C(X ) (or Riemann integrable), a ∈ G;
(F (an Γ))n∈N is called a basic `-step nilsequence;
`-step nilsequence= a uniform limit of basic `-step nilsequences.
1-step nilsequence = almost periodic sequence.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
7 / 17
R
Double correlations ( f · T n g · T 2n h dµ)-Nilsequences.
Examples
2
(e2πi(n α+nβ) )n∈N , (e2πi[nα]nβ · φ(nα))n∈N for some φ ∈ C(T).
Also finite products of such sequences.
All these are particular cases of:
Definition (`-step nilsequences)
X = G/Γ where G is a `-step nilpotent group and Γ is a discrete
cocompact subgroup;
F ∈ C(X ) (or Riemann integrable), a ∈ G;
(F (an Γ))n∈N is called a basic `-step nilsequence;
`-step nilsequence= a uniform limit of basic `-step nilsequences.
1-step nilsequence = almost periodic sequence.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
7 / 17
Structure of multiple correlations (Ti = T i ).
Theorem (Bergelson, Host, Kra 05)
For a given ergodic system (X , µ, T ) and fi ∈ L∞ (µ) we have
Z
f0 · T n f1 · . . . · T `n f` dµ = ast (n) + aer (n), n ∈ N,
where
(ast (n))n∈N is an `-step nilsequence; and
1 PN−1
limN−M→∞ N−M
n=M |aer (n)| = 0.
A similar result holds for polynomial iterates [Leibman 10] and
non-ergodic systems [Leibman 15].
Method of proof relies on the theory of characteristic factors and is
inapplicable for commuting transformations.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
8 / 17
Structure of multiple correlations (Ti = T i ).
Theorem (Bergelson, Host, Kra 05)
For a given ergodic system (X , µ, T ) and fi ∈ L∞ (µ) we have
Z
f0 · T n f1 · . . . · T `n f` dµ = ast (n) + aer (n), n ∈ N,
where
(ast (n))n∈N is an `-step nilsequence; and
1 PN−1
limN−M→∞ N−M
n=M |aer (n)| = 0.
A similar result holds for polynomial iterates [Leibman 10] and
non-ergodic systems [Leibman 15].
Method of proof relies on the theory of characteristic factors and is
inapplicable for commuting transformations.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
8 / 17
Structure of multiple correlations (general T1 , . . . , T` ).
Theorem (F. 15)
Given (X , µ, T1 , . . . , T` ), fi ∈ L∞ (µ), and ε > 0 we have
Z
f0 · T1n f1 · . . . · T`n f` dµ = ast (n) + aer (n), n ∈ N,
where
(ast (n))n∈N is a basic `-step nilsequence; and
1 PN−1
lim supN−M→∞ N−M
n=M |aer (n)| ≤ ε.
R
p (n)
p (n)
Similar results for f0 · T1 1 f1 · . . . · T` ` f` dµ, and when
T1 , . . . , T` generate a nilpotent group.
Proof does not rely on theory of characteristic factors. It only uses
the single transformation theory and mean convergence results.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
9 / 17
Structure of multiple correlations (general T1 , . . . , T` ).
Theorem (F. 15)
Given (X , µ, T1 , . . . , T` ), fi ∈ L∞ (µ), and ε > 0 we have
Z
f0 · T1n f1 · . . . · T`n f` dµ = ast (n) + aer (n), n ∈ N,
where
(ast (n))n∈N is a basic `-step nilsequence; and
1 PN−1
lim supN−M→∞ N−M
n=M |aer (n)| ≤ ε.
R
p (n)
p (n)
Similar results for f0 · T1 1 f1 · . . . · T` ` f` dµ, and when
T1 , . . . , T` generate a nilpotent group.
Proof does not rely on theory of characteristic factors. It only uses
the single transformation theory and mean convergence results.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
9 / 17
Structure of multiple correlations (general T1 , . . . , T` ).
Theorem (F. 15)
Given (X , µ, T1 , . . . , T` ), fi ∈ L∞ (µ), and ε > 0 we have
Z
f0 · T1n f1 · . . . · T`n f` dµ = ast (n) + aer (n), n ∈ N,
where
(ast (n))n∈N is a basic `-step nilsequence; and
1 PN−1
lim supN−M→∞ N−M
n=M |aer (n)| ≤ ε.
R
p (n)
p (n)
Similar results for f0 · T1 1 f1 · . . . · T` ` f` dµ, and when
T1 , . . . , T` generate a nilpotent group.
Proof does not rely on theory of characteristic factors. It only uses
the single transformation theory and mean convergence results.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
9 / 17
Two properties that characterize multiple correlations
1
a ∈ `∞ is `-anti-uniform if there exists C = Ca s.t. ∀ b ∈ `∞
lim sup N−M→∞
N−1
X
1
a(n) · b(n) ≤ C kbkU`+1 .
N −M
n=M
8
kbkU3 = Avr ,s∈N |Avm∈N b(m) · b(m + r ) · b(m + s) · b(m + r + s)|2 .
van der Corput =⇒ multiple correlations are `-anti-uniform.
2
a ∈ `∞ is `-regular if the limit
N−1
X
1
a(n) · ψ(n)
N−M→∞ N − M
lim
n=M
exists for every `-step nilsequence (ψ(n))n∈N .
Convergence (Tao) + trick =⇒ multiple correlations are `-regular.
Trick: `-step nilsequence = multiple correlation + small `∞ -error.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
10 / 17
Two properties that characterize multiple correlations
1
a ∈ `∞ is `-anti-uniform if there exists C = Ca s.t. ∀ b ∈ `∞
lim sup N−M→∞
N−1
X
1
a(n) · b(n) ≤ C kbkU`+1 .
N −M
n=M
8
kbkU3 = Avr ,s∈N |Avm∈N b(m) · b(m + r ) · b(m + s) · b(m + r + s)|2 .
van der Corput =⇒ multiple correlations are `-anti-uniform.
2
a ∈ `∞ is `-regular if the limit
N−1
X
1
a(n) · ψ(n)
N−M→∞ N − M
lim
n=M
exists for every `-step nilsequence (ψ(n))n∈N .
Convergence (Tao) + trick =⇒ multiple correlations are `-regular.
Trick: `-step nilsequence = multiple correlation + small `∞ -error.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
10 / 17
Anti-uniformity + Regularity ⇒ Decomposition
Theorem (F. 15)
If a ∈ `∞ is `-anti-uniform and `-regular, then for every ε > 0
a(n) = ast (n) + aer (n),
n ∈ N,
where
(ast (n))n∈N is a basic `-step nilsequence; and
1 PN−1
lim supN−M→∞ N−M
n=M |aer (n)| ≤ ε.
Key ingredient in the proof an inverse theorem of Host and Kra.
If a(n) = C` (n) we get the decomposition for multiple correlations.
(van der Corput+Vitaly’s PET)+ (Walsh convergence result) =⇒
decomposition for correlation sequences with polynomial iterates.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
11 / 17
Anti-uniformity + Regularity ⇒ Decomposition
Theorem (F. 15)
If a ∈ `∞ is `-anti-uniform and `-regular, then for every ε > 0
a(n) = ast (n) + aer (n),
n ∈ N,
where
(ast (n))n∈N is a basic `-step nilsequence; and
1 PN−1
lim supN−M→∞ N−M
n=M |aer (n)| ≤ ε.
Key ingredient in the proof an inverse theorem of Host and Kra.
If a(n) = C` (n) we get the decomposition for multiple correlations.
(van der Corput+Vitaly’s PET)+ (Walsh convergence result) =⇒
decomposition for correlation sequences with polynomial iterates.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
11 / 17
Convergence criterions
Corollary
Let (wn )n∈N be a bounded sequence. The following are equivalent:
R
1 PN
n
n
n=1 wn · f0 · T1 f1 · . . . · T` f` dµ converge ∀ (X , T1 , . . . , T` ), fi ;
N
P
N
1
n=1 wn · ψ(n) converge ∀ `-step nilsequence (ψ(n))n∈N .
N
Using this result, one can extend weighted ergodic theorems of
Host-Kra, Chu, Eisner-Zorin-Kranich, Eisner, to the case of commuting
transformations.
Problem
Show that the previous result holds for all (wn ) (possibly unbounded).
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
12 / 17
Convergence criterions
Corollary
Let (wn )n∈N be a bounded sequence. The following are equivalent:
R
1 PN
n
n
n=1 wn · f0 · T1 f1 · . . . · T` f` dµ converge ∀ (X , T1 , . . . , T` ), fi ;
N
P
N
1
n=1 wn · ψ(n) converge ∀ `-step nilsequence (ψ(n))n∈N .
N
Using this result, one can extend weighted ergodic theorems of
Host-Kra, Chu, Eisner-Zorin-Kranich, Eisner, to the case of commuting
transformations.
Problem
Show that the previous result holds for all (wn ) (possibly unbounded).
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
12 / 17
Convergence criterions
Corollary
Let (wn )n∈N be a bounded sequence. The following are equivalent:
R
1 PN
n
n
n=1 wn · f0 · T1 f1 · . . . · T` f` dµ converge ∀ (X , T1 , . . . , T` ), fi ;
N
P
N
1
n=1 wn · ψ(n) converge ∀ `-step nilsequence (ψ(n))n∈N .
N
Using this result, one can extend weighted ergodic theorems of
Host-Kra, Chu, Eisner-Zorin-Kranich, Eisner, to the case of commuting
transformations.
Problem
Show that the previous result holds for all (wn ) (possibly unbounded).
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
12 / 17
Convergence criterions
Corollary
Let (wn )n∈N be a bounded sequence. The following are equivalent:
R
1 PN
n
n
n=1 wn · f0 · T1 f1 · . . . · T` f` dµ converge ∀ (X , T1 , . . . , T` ), fi ;
N
P
N
1
n=1 wn · ψ(n) converge ∀ `-step nilsequence (ψ(n))n∈N .
N
Using this result, one can extend weighted ergodic theorems of
Host-Kra, Chu, Eisner-Zorin-Kranich, Eisner, to the case of commuting
transformations.
Problem
Show that the previous result holds for all (wn ) (possibly unbounded).
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
12 / 17
Applications: Results for sets coming from arithmetic
Sa,b = {n ∈ N : number of distinct prime factors ≡ a mod b}
Theorem (F., Host 2015)
For every a, b, c, system (X , µ, T1 , . . . , T` ), and µ(A) > 0, we have
µ(A ∩ T1−n A ∩ · · · ∩ T`−n A) > 0 for some n ∈ Sa,b + c.
Theorem (F., Host 2015)
If (X , µ, T1 , . . . , T` ) is a system and f1 , . . . , f` ∈ L∞ (µ), then the
following averages converge in L2 (µ)
1
N
X
T1n f1 · · · T`n f` .
n∈Sa,b ∩[N]
Similar results holds for polynomial iterates.
Key: An ergodic theorem with weights given by multiplicative functions.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
13 / 17
Applications: Results for sets coming from arithmetic
Sa,b = {n ∈ N : number of distinct prime factors ≡ a mod b}
Theorem (F., Host 2015)
For every a, b, c, system (X , µ, T1 , . . . , T` ), and µ(A) > 0, we have
µ(A ∩ T1−n A ∩ · · · ∩ T`−n A) > 0 for some n ∈ Sa,b + c.
Theorem (F., Host 2015)
If (X , µ, T1 , . . . , T` ) is a system and f1 , . . . , f` ∈ L∞ (µ), then the
following averages converge in L2 (µ)
1
N
X
T1n f1 · · · T`n f` .
n∈Sa,b ∩[N]
Similar results holds for polynomial iterates.
Key: An ergodic theorem with weights given by multiplicative functions.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
13 / 17
Applications: Results for sets coming from arithmetic
Sa,b = {n ∈ N : number of distinct prime factors ≡ a mod b}
Theorem (F., Host 2015)
For every a, b, c, system (X , µ, T1 , . . . , T` ), and µ(A) > 0, we have
µ(A ∩ T1−n A ∩ · · · ∩ T`−n A) > 0 for some n ∈ Sa,b + c.
Theorem (F., Host 2015)
If (X , µ, T1 , . . . , T` ) is a system and f1 , . . . , f` ∈ L∞ (µ), then the
following averages converge in L2 (µ)
1
N
X
T1n f1 · · · T`n f` .
n∈Sa,b ∩[N]
Similar results holds for polynomial iterates.
Key: An ergodic theorem with weights given by multiplicative functions.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
13 / 17
Connection with multiplicative functions
Both results follow from the identity
N
N
1X n
b X
n
n
1Sa,b (n) · T1 f1 · · · T` f` =
T1 f1 · · · T`n f` + oN→∞ (1).
N
N
n=1
(1)
n=1
If ζ is a root of unity of order b we let φb (n) = ζ |distinct prime factors of n| .
Then φb (n) = ζ a ⇐⇒ n ∈ Sa,b , hence
b−1
1Sa,b (n) =
1 X −aj j
ζ φb (n).
b
j=0
Identity (1) follows from
N
1X j
φb (n) · T1n f1 · · · T`n f` → 0,
N
j = 1, . . . , b − 1.
n=1
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
14 / 17
Connection with multiplicative functions
Both results follow from the identity
N
N
1X n
b X
n
n
1Sa,b (n) · T1 f1 · · · T` f` =
T1 f1 · · · T`n f` + oN→∞ (1).
N
N
n=1
(1)
n=1
If ζ is a root of unity of order b we let φb (n) = ζ |distinct prime factors of n| .
Then φb (n) = ζ a ⇐⇒ n ∈ Sa,b , hence
b−1
1Sa,b (n) =
1 X −aj j
ζ φb (n).
b
j=0
Identity (1) follows from
N
1X j
φb (n) · T1n f1 · · · T`n f` → 0,
N
j = 1, . . . , b − 1.
n=1
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
14 / 17
Connection with multiplicative functions
Both results follow from the identity
N
N
1X n
b X
n
n
1Sa,b (n) · T1 f1 · · · T` f` =
T1 f1 · · · T`n f` + oN→∞ (1).
N
N
n=1
(1)
n=1
If ζ is a root of unity of order b we let φb (n) = ζ |distinct prime factors of n| .
Then φb (n) = ζ a ⇐⇒ n ∈ Sa,b , hence
b−1
1Sa,b (n) =
1 X −aj j
ζ φb (n).
b
j=0
Identity (1) follows from
N
1X j
φb (n) · T1n f1 · · · T`n f` → 0,
N
j = 1, . . . , b − 1.
n=1
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
14 / 17
Multiplicative functions
Definition (Multiplicative functions)
M = {φ : N → C : φ(mn) = φ(m)φ(n) whenever (m, n) = 1}
Not all multiplicative functions have P
a mean value.
1
it
it
it
For example φ(n) = n does not, N N
n=1 n ∼ N /(1 + it).
Definition (Multiplicative functions with convergent means)
Mconv is the set of bounded multiplicative functions φ such that
N
1X
φ(an + b) exists for all a, b ∈ N.
N→∞ N
lim
n=1
If all these mean values are 0 we call φ aperiodic.
It can be shown that φjb is aperiodic for j = 1, . . . , b − 1. Also, the
Möbius and Liouville functions are aperiodic.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
15 / 17
Multiplicative functions
Definition (Multiplicative functions)
M = {φ : N → C : φ(mn) = φ(m)φ(n) whenever (m, n) = 1}
Not all multiplicative functions have P
a mean value.
1
it
it
it
For example φ(n) = n does not, N N
n=1 n ∼ N /(1 + it).
Definition (Multiplicative functions with convergent means)
Mconv is the set of bounded multiplicative functions φ such that
N
1X
φ(an + b) exists for all a, b ∈ N.
N→∞ N
lim
n=1
If all these mean values are 0 we call φ aperiodic.
It can be shown that φjb is aperiodic for j = 1, . . . , b − 1. Also, the
Möbius and Liouville functions are aperiodic.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
15 / 17
Multiplicative functions
Definition (Multiplicative functions)
M = {φ : N → C : φ(mn) = φ(m)φ(n) whenever (m, n) = 1}
Not all multiplicative functions have P
a mean value.
1
it
it
it
For example φ(n) = n does not, N N
n=1 n ∼ N /(1 + it).
Definition (Multiplicative functions with convergent means)
Mconv is the set of bounded multiplicative functions φ such that
N
1X
φ(an + b) exists for all a, b ∈ N.
N→∞ N
lim
n=1
If all these mean values are 0 we call φ aperiodic.
It can be shown that φjb is aperiodic for j = 1, . . . , b − 1. Also, the
Möbius and Liouville functions are aperiodic.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
15 / 17
An ergodic theorem with multiplicative weights
Theorem (F., Host 2015)
Let φ ∈ Mconv . If (X , µ, T1 , . . . , T` ) is a system and f1 , . . . , f` ∈ L∞ (µ),
then the averages
N
1X
φ(n) · T1n f1 · · · T`n f`
N
n=1
converge in L2 (µ). Furthermore, they converge to 0 if φ is aperiodic.
φ ∈ Mconv is necessary for convergence even when ` = 1.
Similar statement for polynomial iterates.
Second part applies to φjb for j = 1, . . . , b − 1.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
16 / 17
An ergodic theorem with multiplicative weights
Theorem (F., Host 2015)
Let φ ∈ Mconv . If (X , µ, T1 , . . . , T` ) is a system and f1 , . . . , f` ∈ L∞ (µ),
then the averages
N
1X
φ(n) · T1n f1 · · · T`n f`
N
n=1
converge in L2 (µ). Furthermore, they converge to 0 if φ is aperiodic.
φ ∈ Mconv is necessary for convergence even when ` = 1.
Similar statement for polynomial iterates.
Second part applies to φjb for j = 1, . . . , b − 1.
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
16 / 17
Sketch of proof
Using the decomposition result weak convergence follows from:
Theorem (F., Host 2014)
Let φ ∈ Mconv . Then for every nilsequence (ψ(n)) the limit
N
1X
φ(n) · ψ(n)
N→∞ N
lim
n=1
exists. It is 0 if φ is aperiodic.
For mean convergence: If AN :=
1
N
PN
n=1 φ(n)
· T1n f1 · . . . · T`n f` , then
kAN − AM k2L2 (µ) = weighted average of 2-variable correlation sequences.
It can be shown [F., Host 2015] that
2-variable correlation sequence= (2-variable nilsequence) + (small error)
THANK YOU!
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
17 / 17
Sketch of proof
Using the decomposition result weak convergence follows from:
Theorem (F., Host 2014)
Let φ ∈ Mconv . Then for every nilsequence (ψ(n)) the limit
N
1X
φ(n) · ψ(n)
N→∞ N
lim
n=1
exists. It is 0 if φ is aperiodic.
For mean convergence: If AN :=
1
N
PN
n=1 φ(n)
· T1n f1 · . . . · T`n f` , then
kAN − AM k2L2 (µ) = weighted average of 2-variable correlation sequences.
It can be shown [F., Host 2015] that
2-variable correlation sequence= (2-variable nilsequence) + (small error)
THANK YOU!
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
17 / 17
Sketch of proof
Using the decomposition result weak convergence follows from:
Theorem (F., Host 2014)
Let φ ∈ Mconv . Then for every nilsequence (ψ(n)) the limit
N
1X
φ(n) · ψ(n)
N→∞ N
lim
n=1
exists. It is 0 if φ is aperiodic.
For mean convergence: If AN :=
1
N
PN
n=1 φ(n)
· T1n f1 · . . . · T`n f` , then
kAN − AM k2L2 (µ) = weighted average of 2-variable correlation sequences.
It can be shown [F., Host 2015] that
2-variable correlation sequence= (2-variable nilsequence) + (small error)
THANK YOU!
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
17 / 17
Sketch of proof
Using the decomposition result weak convergence follows from:
Theorem (F., Host 2014)
Let φ ∈ Mconv . Then for every nilsequence (ψ(n)) the limit
N
1X
φ(n) · ψ(n)
N→∞ N
lim
n=1
exists. It is 0 if φ is aperiodic.
For mean convergence: If AN :=
1
N
PN
n=1 φ(n)
· T1n f1 · . . . · T`n f` , then
kAN − AM k2L2 (µ) = weighted average of 2-variable correlation sequences.
It can be shown [F., Host 2015] that
2-variable correlation sequence= (2-variable nilsequence) + (small error)
THANK YOU!
Nikos Frantzikinakis (U. of Crete)
Multiple correlation sequences
Bedlewo, November 2015
17 / 17