almost-prime times in horospherical flowslior/sem/slides/2020.05.28...2020/05/28 · i margulis,...
TRANSCRIPT
![Page 1: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/1.jpg)
Almost-Prime Times in Horospherical FlowsWest Coast Dynamics Seminar
Taylor McAdam
Yale University
May 28, 2020
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 2: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/2.jpg)
Homogeneous Dynamics
I G, a Lie groupI Γ≤ G, a lattice (discrete, finite covolume subgroup)I X = Γ\G, space of interestI H ≤ G, a closed subgroupI Dynamics: H y X by right translations
Possible questions:I Given x ∈ X , what does the orbit xH look like?I What does a typical orbit look like?I What H-invariant/ergodic measures are supported on this space?
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 3: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/3.jpg)
Homogeneous Dynamics
I G, a Lie groupI Γ≤ G, a lattice (discrete, finite covolume subgroup)I X = Γ\G, space of interestI H ≤ G, a closed subgroupI Dynamics: H y X by right translations
Possible questions:I Given x ∈ X , what does the orbit xH look like?I What does a typical orbit look like?I What H-invariant/ergodic measures are supported on this space?
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 4: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/4.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 5: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/5.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 6: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/6.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 7: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/7.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 8: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/8.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 9: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/9.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 10: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/10.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 11: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/11.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 12: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/12.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 13: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/13.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 14: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/14.jpg)
Example: Linear Flows on the Torus
G = R2, Γ = Z2, X = T2, H = {tv t ∈ R} for some v ∈ R2
I If v has rational slope, then every orbit is periodic.I If v has irrational slope, then every orbit is dense.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 15: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/15.jpg)
The Space of Lattices
I G = SL2(R)I Γ = SL2(Z)I G yH2 := {z = x+ iy ∈ C y > 0} by Mobius transformations:
g =
(a bc d
): z 7→ az+b
cz+d
I G y T 1H2 by g : (z,v) 7→ (g(z),Dgv) with StabG(z) = {±I}I PSL2(R)∼= T 1H2
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Space of Lattices
PSL2(Z)\PSL2(R)
dxdydθ
y2
i
−1/2 1/2
SL2(Z)g←→ Z2g
1
G = SLn(R), Γ = SLn(Z), Γ\G∼= {lattices in Rn of covolume 1}
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Space of Lattices
PSL2(Z)\PSL2(R)
dxdydθ
y2
i
−1/2 1/2
SL2(Z)g←→ Z2g
1
G = SLn(R), Γ = SLn(Z), Γ\G∼= {lattices in Rn of covolume 1}
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Space of Lattices
PSL2(R)∼= T1H2
−1/2 1/2
SL2(Z)g←→ Z2g
1
G = SLn(R), Γ = SLn(Z), Γ\G∼= {lattices in Rn of covolume 1}
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Space of Lattices
PSL2(R)∼= T1H2
−1/2 1/2
SL2(Z)g←→ Z2g
1
G = SLn(R), Γ = SLn(Z), Γ\G∼= {lattices in Rn of covolume 1}
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 20: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/20.jpg)
Subgroup Actions
geodesic flow horocycle flow
A =
{at =
(et/2 00 e−t/2
)}t∈R
U =
{ut =
(1 t0 1
)}t∈R
PSL2(R)∼= T1H2
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Subgroup Actions
geodesic flow horocycle flow
A =
{at =
(et/2 00 e−t/2
)}t∈R
U =
{ut =
(1 t0 1
)}t∈R
PSL2(R)∼= T1H2
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 22: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/22.jpg)
Subgroup Actions
geodesic flow horocycle flow
A =
{at =
(et/2 00 e−t/2
)}t∈R
U =
{ut =
(1 t0 1
)}t∈R
PSL2(R)∼= T1H2
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 23: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/23.jpg)
Horospherical Subgroups
Note: a−1t usat =
(1 se−t
0 1
)→(
1 00 1
)as t→ ∞
DefinitionA subgroup H ≤ G is called horospherical if there exists g ∈ G suchthat
H = {h ∈ G g−nhgn→ e as n→ ∞}.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Subgroups
Note: a−1t usat =
(1 se−t
0 1
)→(
1 00 1
)as t→ ∞
DefinitionA subgroup H ≤ G is called horospherical if there exists g ∈ G suchthat
H = {h ∈ G g−nhgn→ e as n→ ∞}.
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 25: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/25.jpg)
Horospherical Subgroups
Fact: horospherical =⇒6⇐= unipotent
Example (Heisenberg group)1 x y
1 z1
x,y,z ∈ R
with respect to, e.g.,
21
12
Example1 t t2/2
1 t1
t ∈ R
is NOT horospherical
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Subgroups
Fact: horospherical =⇒6⇐= unipotent
Example (Heisenberg group)1 x y
1 z1
x,y,z ∈ R
with respect to, e.g.,
21
12
Example1 t t2/2
1 t1
t ∈ R
is NOT horospherical
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Subgroups
Fact: horospherical =⇒6⇐= unipotent
Example (Heisenberg group)1 x y
1 z1
x,y,z ∈ R
with respect to, e.g.,
21
12
Example1 t t2/2
1 t1
t ∈ R
is NOT horospherical
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 28: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/28.jpg)
Equidistribution
Roughly speaking, a subset of X equidistributes respect to a measureµ if it spends the expected amount of time in measurable subsets.
Example
A sequence {xn}n∈N ⊂ X equidistributes with respect to µ if
1N
N
∑n=1
f (xn)→∫
Xf dµ
for all f ∈C∞c (X).
Say equidistribution is effective if the rate of convergence is known.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Equidistribution
Roughly speaking, a subset of X equidistributes respect to a measureµ if it spends the expected amount of time in measurable subsets.
Example
A sequence {xn}n∈N ⊂ X equidistributes with respect to µ if
1N
N
∑n=1
f (xn)→∫
Xf dµ
for all f ∈C∞c (X).
Say equidistribution is effective if the rate of convergence is known.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Equidistribution
Roughly speaking, a subset of X equidistributes respect to a measureµ if it spends the expected amount of time in measurable subsets.
Example
A path {x(t)}t∈R+ ⊂ X equidistributes with respect to µ if
1T
∫ T
0f (x(t))dt→
∫X
f dµ
for all f ∈C∞c (X).
Say equidistribution is effective if the rate of convergence is known.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Rigidity of Horospherical Actions
TheoremLet H ≤ G be horospherical. For any x ∈ X, there exists a closed,connected subgroup H ≤ L≤ G such that xH = xL and such that xLsupports an L-invariant probability measure µx with respect to whichthe H-orbit of x equidistributes.
I Hedlund, Furstenberg (SL2)I Burger (SL2, Γ cocompact, effective w/ polynomial rate)I Veech, Ellis-Perrizo (general horospherical, Γ cocompact)I Margulis, Dani, Dani-Margulis (quantitative nondivergence)I Dani (above theorem)I Strombergsson, Flaminio-Forni
(SL2, Γ non-uniform, effective w/ polynomial rate depending onbasepoint)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Rigidity of Horospherical Actions
TheoremLet H ≤ G be horospherical. For any x ∈ X, there exists a closed,connected subgroup H ≤ L≤ G such that xH = xL and such that xLsupports an L-invariant probability measure µx with respect to whichthe H-orbit of x equidistributes.
I Hedlund, Furstenberg (SL2)I Burger (SL2, Γ cocompact, effective w/ polynomial rate)I Veech, Ellis-Perrizo (general horospherical, Γ cocompact)I Margulis, Dani, Dani-Margulis (quantitative nondivergence)I Dani (above theorem)I Strombergsson, Flaminio-Forni
(SL2, Γ non-uniform, effective w/ polynomial rate depending onbasepoint)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Qualitative Equidistribution
Theorem (Dani)For every x = Γg ∈ X, either
1|BT |
∫BT
f (xu)du−−−→T→∞
∫X
f dm ∀ f ∈C∞c (X) (1)
or there is a proper, nontrivial rational subspace W ⊂ Rn such thatWg is U-invariant.
I du Haar measure on UI dm pushforward of Haar measure on G to XI BT = alogT BU
1 a−1logT expanding Følner sets
I If x satisfies (1), call it generic.(Birkhoff’s Theorem =⇒ almost every x is generic.)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution
Theorem (M.)There exists γ > 0 such that for every x = Γg ∈ X and T > R largeenough, either:∣∣∣∣ 1
|BT |
∫BT
f (xu)du−∫
Xf dm
∣∣∣∣� f R−γ ∀ f ∈C∞c (X) (2a)
or
∃ j ∈ {1, · · · ,n−1} and w ∈ Λj(Zn)\{0} such that
||wg0u||< R ∀u ∈ BT .(2b)
I If x satisfies (2a) for fixed R and all large T , call it R-generic.Note: x is generic ⇐⇒ x is R-generic for all R > 0.
I Condition (2b) says that there is a rational subspace W ∈Rn suchthat Wg is R-almost invariant when flowed up to time T .
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution
Theorem (M.)There exists γ > 0 such that for every x = Γg ∈ X and T > R largeenough, either:∣∣∣∣ 1
|BT |
∫BT
f (xu)du−∫
Xf dm
∣∣∣∣� f R−γ ∀ f ∈C∞c (X) (2a)
or
∃ j ∈ {1, · · · ,n−1} and w ∈ Λj(Zn)\{0} such that
||wg0u||< R ∀u ∈ BT .(2b)
I If x satisfies (2a) for fixed R and all large T , call it R-generic.Note: x is generic ⇐⇒ x is R-generic for all R > 0.
I Condition (2b) says that there is a rational subspace W ∈Rn suchthat Wg is R-almost invariant when flowed up to time T .
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Effective Equidistribution
Theorem (M.)There exists γ > 0 such that for every x = Γg ∈ X and T > R largeenough, either:∣∣∣∣ 1
|BT |
∫BT
f (xu)du−∫
Xf dm
∣∣∣∣� f R−γ ∀ f ∈C∞c (X) (2a)
or
∃ j ∈ {1, · · · ,n−1} and w ∈ Λj(Zn)\{0} such that
||wg0u||< R ∀u ∈ BT .(2b)
I If x satisfies (2a) for fixed R and all large T , call it R-generic.Note: x is generic ⇐⇒ x is R-generic for all R > 0.
I Condition (2b) says that there is a rational subspace W ∈Rn suchthat Wg is R-almost invariant when flowed up to time T .
Taylor McAdam Almost-Prime Times in Horospherical Flows
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But Why?
Why do we want effective results?
Taylor McAdam Almost-Prime Times in Horospherical Flows
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But Why?
Why do we want effective results?
I Applications in number theory often require effective rates.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Mobius Disjointness
Recall: the Mobius function
µ(n) =
{0 if n is not squarefree(−1)k if n is the product of k distinct primes
Conjecture (Sarnak)
1N ∑
n≤Nµ(n) f (T nx)→ 0
for any:I X compact metric spaceI x ∈ XI T : X → X continuous, zero topological entropyI f ∈C(X)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Mobius Disjointness
Recall: the Mobius function
µ(n) =
{0 if n is not squarefree(−1)k if n is the product of k distinct primes
Conjecture (Sarnak)
1N ∑
n≤Nµ(n) f (T nx)→ 0
for any:I X compact metric spaceI x ∈ XI T : X → X continuous, zero topological entropyI f ∈C(X)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Mobius Disjointness
Partial results:I Vinogradov/Davenport (circle rotations/translations on a
compact group–effective)I Green-Tao (nilflows—effective)I Bourgain-Sarnak-Ziegler/Peckner (unipotent flows on
homogeneous spaces—not effective)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Equidistribution of Primes
Conjecture (Margulis)
Let {ut}t∈R be a unipotent flow on a homogeneous space X. If{xut t ∈ R} equidistributes in X, then so does {xup p is prime}.
Theorem (Bourgain)
For any measurable dynamical system (X ,B,µ,T ) and f ∈ L2(X ,µ),the ergodic averages over primes
1π(N) ∑
p≤Np prime
f (T px)
converge for µ-a.e x ∈ X.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Equidistribution of Primes
Conjecture (Margulis)
Let {ut}t∈R be a unipotent flow on a homogeneous space X. If{xut t ∈ R} equidistributes in X, then so does {xup p is prime}.
Theorem (Bourgain)
For any measurable dynamical system (X ,B,µ,T ) and f ∈ L2(X ,µ),the ergodic averages over primes
1π(N) ∑
p≤Np prime
f (T px)
converge for µ-a.e x ∈ X.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Horocycle Flow at Almost-Prime Times
DefinitionAn integer is called almost-prime if it has fewer than a fixed numberof prime factors.
Theorem (Sarnak-Ubis)
There exists ` ∈ N such that for any generic x ∈ SL2(Z)\SL2(R), theset
{xu(k) k ∈ Z has fewer than ` prime factors}
is dense in SL2(Z)\SL2(R).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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The Horocycle Flow at Almost-Prime Times
DefinitionAn integer is called almost-prime if it has fewer than a fixed numberof prime factors.
Theorem (Sarnak-Ubis)
There exists ` ∈ N such that for any generic x ∈ SL2(Z)\SL2(R), theset
{xu(k) k ∈ Z has fewer than ` prime factors}
is dense in SL2(Z)\SL2(R).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Flows at Almost-Prime Times
Let G = SLn(R), Γ≤ G a lattice, and u(t) a d-dimensionalhorospherical flow on X = Γ\G. Define
A`(x) = {xu(k1,k2, · · · ,kd) | ki ∈ Z has fewer than ` prime factors}.
Theorem (M.)
1. If Γ is cocompact, then there exists `= `(n,d,Γ) such that forany x ∈ X, the set A`(x) is dense in X.
2. If Γ = SLn(Z) and x = Γg ∈ X satisfies a Diophantine propertywith parameter δ, then there exists `= `(n,d,δ) such that A`(x)is dense in X.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Flows at Almost-Prime Times
Let G = SLn(R), Γ≤ G a lattice, and u(t) a d-dimensionalhorospherical flow on X = Γ\G. Define
A`(x) = {xu(k1,k2, · · · ,kd) | ki ∈ Z has fewer than ` prime factors}.
Theorem (M.)
1. If Γ is cocompact, then there exists `= `(n,d,Γ) such that forany x ∈ X, the set A`(x) is dense in X.
2. If Γ = SLn(Z) and x = Γg ∈ X satisfies a Diophantine propertywith parameter δ, then there exists `= `(n,d,δ) such that A`(x)is dense in X.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Horospherical Flows at Almost-Prime Times
Let G = SLn(R), Γ≤ G a lattice, and u(t) a d-dimensionalhorospherical flow on X = Γ\G. Define
A`(x) = {xu(k1,k2, · · · ,kd) | ki ∈ Z has fewer than ` prime factors}.
Theorem (M.)
1. If Γ is cocompact, then there exists `= `(n,d,Γ) such that forany x ∈ X, the set A`(x) is dense in X.
2. If Γ = SLn(Z) and x = Γg ∈ X satisfies a Diophantine propertywith parameter δ, then there exists `= `(n,d,δ) such that A`(x)is dense in X.
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Questions?
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Almost-Primes in Horospherical Flows
Proof Idea:
1. Prove effective equidistribution of the continuous horosphericalflow
2. Use this to prove effective equidistribution of arithmeticprogressions of times
3. Apply sieve methods to deduce a statement about almost-primes
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
Proof Idea: Margulis’s thickening method
xu(T )
xxalogT u(s)a−1
logT
xu(1)
xxu(s)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
Proof Idea: Margulis’s thickening method
xu(T )
xysa−1
logT
xu(1)
xxu(s)
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
Effective mixing of the A-action:
Theorem (Howe-Moore, Kleinbock-Margulis)Let Γ be cocompact. There exists γ > 0 such that for any x ∈ X andf ,g ∈C∞
c (X),∣∣∣∣∫Xf (xat)g(x)dm−
∫X
f dm∫
Xgdm
∣∣∣∣� f ,g e−γt .
Note:
1|BT |
∫BT
f (xu)du =∫
B1
f (xalogT ua−1logT )du
=∫
UχB1(u) f (xalogT ua−1
logT )du
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
Effective mixing of the A-action:
Theorem (Howe-Moore, Kleinbock-Margulis)Let Γ be cocompact. There exists γ > 0 such that for any x ∈ X andf ,g ∈C∞
c (X),∣∣∣∣∫Xf (xat)g(x)dm−
∫X
f dm∫
Xgdm
∣∣∣∣� f ,g e−γt .
Note:
1|BT |
∫BT
f (xu)du =∫
B1
f (xalogT ua−1logT )du
=∫
UχB1(u) f (xalogT ua−1
logT )du
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of the Continuous Flow
∫U
χB1(u) f (xalogT ua−1logT )du
Problems:
1. χB1 not smoothI Convolve with a smooth approximation to the identity
2. Integral over U , not XI Thicken to get integral in G, project to X (need to make sure it
injects)3. Moving basepoint
I Quantitative nondivergence (Dani-Margulis) =⇒ can get a goodradius of convergence for all but a small proportion of u ∈ B1
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Theorem (M.)
Let u(t1, · · · , td) be an abelian horospherical flow. There exists β > 0such that if x ∈ X satisfies (2a) for T > R large enough, then for any1≤ K ≤ T we have∣∣∣∣∣∣∣
Kd
T d ∑k∈Zd
Kk∈BT
f (xu(Kk))−∫
Xf dm
∣∣∣∣∣∣∣� f R−βKd/(d+1)S( f ).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Proof Idea: Venkatesh’s van der Corput method
For simplicity, assume G = SL2(R),∫
f dm = 0.
LetEK,T ( f ) =
KT ∑
k∈Z0≤Kk<T
f (xu(Kk))
be the average over the set:
x xu(T )xu(K) xu(2K)
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Effective Equidistribution of Arithmetic Progressions
Define new function for 1 < H < T :
fH(x) =1H
H
∑`=0
f (xu(K`))
Note: EK,T ( fH) is close to EK,T ( f ):
1H
0
0
T
H
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Define new function for 1 < H < T :
fH(x) =1H
H
∑`=0
f (xu(K`))
Note: EK,T ( fH) is close to EK,T ( f ):
1H
0
0
T
H
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Thicken the discrete set in U by δ > 0:
x xu(T )xu(K) xu(2K)
Let EK,T,δ be the ergodic average over this set.
Note: By uniform continuity, EK,T,δ( fH) is close to EK,T ( fH).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Thicken the discrete set in U by δ > 0:
x xu(T )xu(K) xu(2K)
Let EK,T,δ be the ergodic average over this set.
Note: By uniform continuity, EK,T,δ( fH) is close to EK,T ( fH).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Note:
EK,T,δ( fH)2� K
δH2
H
∑`1=0
H
∑`2=0
1T
∫ T
0f (xu(s)u(K`1)) f (xu(s)u(K`2))ds
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Note:
EK,T,δ( fH)2� K
δH2
H
∑`1=0
H
∑`2=0
1T
∫ T
0f (xu(s)u(K`1)) f (xu(s)u(K`2))ds
↓ effective equidistribution
� KδH2
H
∑`1=0
H
∑`2=0〈u(K(`1− `2)) · f , f 〉L2(X)+ error
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
Note:
EK,T,δ( fH)2� K
δH2
H
∑`1=0
H
∑`2=0
1T
∫ T
0f (xu(s)u(K`1)) f (xu(s)u(K`2))ds
↓ effective equidistribution
� KδH2
H
∑`1=0
H
∑`2=0〈u(K(`1− `2)) · f , f 〉L2(X)+ error
↓ bounds on matrix coefficients
� KδH2
H
∑`1=0
H
∑`2=0
(1+K|`1− `2|)−aS( f )2 + error
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
KδH2
H
∑`1=0
H
∑`2=0
(1+K|`1− `2|)−aS( f )2
H
H`1 = `2
I Choose H, δ to optimize the various error terms
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Effective Equidistribution of Arithmetic Progressions
KδH2
H
∑`1=0
H
∑`2=0
(1+K|`1− `2|)−aS( f )2
H
H`1 = `2
I Choose H, δ to optimize the various error terms
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Sieving
To sieve the orbits for almost-primes, need control over averagesalong arithmetic progressions—this is exactly what the last theoremtells us.
For f ∈C∞c (X) and T large enough,
(logT )d
T d ∑k∈BT
(k1···kd ,P)=1
f (xu(k))�α
∫f dm
where P is the product of primes less than T α.
Note: The lower bound implies the result for integer points with fewerthan 1/α prime factors (consider f a bump function on any small set).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Sieving
To sieve the orbits for almost-primes, need control over averagesalong arithmetic progressions—this is exactly what the last theoremtells us.
For f ∈C∞c (X) and T large enough,
(logT )d
T d ∑k∈BT
(k1···kd ,P)=1
f (xu(k))�α
∫f dm
where P is the product of primes less than T α.
Note: The lower bound implies the result for integer points with fewerthan 1/α prime factors (consider f a bump function on any small set).
Taylor McAdam Almost-Prime Times in Horospherical Flows
![Page 74: Almost-Prime Times in Horospherical Flowslior/sem/slides/2020.05.28...2020/05/28 · I Margulis, Dani, Dani-Margulis (quantitative nondivergence) I Dani (above theorem) I Strombergsson,](https://reader034.vdocuments.net/reader034/viewer/2022051908/5ffb1c4d37036565924af6fb/html5/thumbnails/74.jpg)
Sieving
To sieve the orbits for almost-primes, need control over averagesalong arithmetic progressions—this is exactly what the last theoremtells us.
For f ∈C∞c (X) and T large enough,
(logT )d
T d ∑k∈BT
(k1···kd ,P)=1
f (xu(k))�α
∫f dm
where P is the product of primes less than T α.
Note: The lower bound implies the result for integer points with fewerthan 1/α prime factors (consider f a bump function on any small set).
Taylor McAdam Almost-Prime Times in Horospherical Flows
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Thank you!
Taylor McAdam Almost-Prime Times in Horospherical Flows