Refinements of Vinogradov's Mean Value Theorem...The case = 0 is Vinogradov’s Mean Value Theorem n...

Refinements of Vinogradov’s Mean ValueTheorem

Ciprian Demeter, IU Bloomington

Madison, May 2019

Ciprian Demeter, IU Bloomington Refinements of Vinogradov’s Mean Value Theorem

Fourier projection of F : Rn → C: PSF (x) =

∫SF (ξ)e(x · ξ)dξ.

Definition (Abstract lq(Lp) decoupling)

Let p, q ≥ 2. For a family S consisting of (possibly infinitely many)pairwise disjoint sets Si ⊂ Rn, let Dec(S, p, q) be the smallestconstant for which the inequality

‖F‖Lp(Rn) ≤ Dec(S, p, q)|S|12− 1

q (∑i

‖PSiF‖qLp(Rn))1/q

holds for each F with Fourier transform supported on ∪Si .

It is immediate that

Dec(S, p, q1) ≤ Dec(S, p, q2), if q1 ≥ q2 (Holder)

Dec(S, 2, q) = 1 (L2 orthogonality)

Dec(S, p, q) ≤ |S|12 (triangle ineq. plus Holder).


Theorem (Littlewood–Paley Theorem implies l2(Lp) decoupling)

Let RLP be the collection of all rectangular boxes R = I1 × . . .× Inin Rn, with each Ii of the form [2ki , 2ki+1] or [−2ki+1,−2ki ] forsome ki ∈ Z. Then for each 1 < p <∞ and each F ∈ Lp(Rn)

‖F‖Lp(Rn) ∼ ‖(∑

R∈RLP

|PRF |2)12 ‖Lp(Rn).

In particular, if p ≥ 2 we have (due to Minkowski’s ineq.)

‖F‖Lp(Rn) . (∑

R∈RLP

‖PRF‖2Lp(Rn))

12 .


We will be interested in decoupling the δ-neighborhood of a curvedmanifold (mostly curves and hypersurfaces) into sets θ

The partition ΘM(δ)


Problem (lq decoupling for manifolds)

Let M be a d-dimensional manifold in Rn

M =Mψ = {(ξ, ψ(ξ)) : ξ ∈ U}.

For each δ ∈ (0, 1), let NM(δ) ⊂ Rn be a set containing theδ-neighborhood of M. Let also ΘM(δ) be a partition of NM(δ)into sets of the form

θ = (τ × Rn−d) ∩ NM(δ),

where τ ranges over a collection PM(δ) of pairwise disjoint almostrectangular boxes in Rd .Given q ≥ 2, find the range 2 ≤ p ≤ pc(q) such that

Dec(ΘM(δ), p, q) .ε δ−ε. (1)

Curvature will play a crucial role


Partition ΘP2(δ) of the δ-neighborhood of the paraboloid intoδ1/2 × δ1/2 × δ boxes θ

Theorem (Bourgain, D., 2014)

The sharp range for l2 decoupling for any hypersurface M⊂ Rn

with positive curvatures is 2 ≤ p ≤ pc := 2(n+1)n−1

Dec(ΘM(δ), p, 2) .ε δ−ε.


Partition ΘΓ3(δ) of the nonisotropic neighborhood of themoment curve

Γ3 : (ξ, ξ2, ξ3), 0 ≤ ξ ≤ 1

into δ1/2 × δ1/3 × δ boxes θ

Theorem (Bourgain, D., Guth 2015)

The sharp range for l2 decoupling for the moment curve Γn ⊂ Rn

is 2 ≤ p ≤ pc := n(n + 1)

Dec(ΘΓn(δ), p, 2) .ε δ−ε.


Flatness is an the enemy of decoupling

Theorem (Doubling exponent for flat decoupling)

Let T be a tube in Rn. Consider a partition TN of T into Nshorter congruent tubes θ. Then for each p, q ≥ 2

Dec(TN , p, q) & N12− 1

q .

In other words, for some F

‖F‖p ∼ N2×( 12− 1

q)(∑θ

‖PθF‖qp)1/q


Partition ΘH2(δ) of the δ-neighborhood of the hyperbolicparaboloid into δ1/2 × δ1/2 × δ boxes θ

Since H2 contains (at least) one line, Dec(ΘM(δ), p, 2) ≥ δ−εpwith εp > 0 if p > 2. So no l2 decoupling, however...


The sharp range for lp decoupling for all surfaces M⊂ R3 withnonzero Gaussian curvature is 2 ≤ p ≤ pc := 4

Dec(ΘM(δ), p, p) .ε δ−ε.


The zero curvature case: the conePartition Θplanks

Co2 (δ) of the δ-neighborhood of the cone into

δ1/2 × 1× δ planks θ


The sharp range for l2 decoupling for the cone Co2 ⊂ R3 is2 ≤ p ≤ pc := 6

Dec(ΘplanksCo2 (δ), p, 2) .ε δ

−ε.


Partition ΘboxesCo2 (δ) of the δ-neighborhood of the cone into

δ1/2 × δ1/2 × δ boxes γ

Conjecture (Bourgain, D., 2014)

The sharp range for lp decoupling for the partition of the cone intoboxes γ is 2 ≤ p ≤ pc := 4

Dec(ΘboxesCo2 (δ), p, p) .ε δ

−ε.


Theorem (Decoupling implies reverse Holder’s inequality)

For each θ ∈ ΘM(δ), let ξθ ∈M∩ θ.Assume either that Dec(ΘM(δ), p, 2) .ε δ

−ε or thatDec(ΘM(δ), p, q) .ε δ

−ε (for some q > 2) and that |aθ| isessentially constant. Then for each ball BR ⊂ Rn with radiusR ≥ δ−1 we have

(1

|BR |

∫BR

|∑θ

aθe(ξθ · x)|pdx)1p .ε δ

−ε‖aθ‖l2 .


The work of Huxley and more recently of Bourgain and Watt(averages of the zeta function on short subintervals of the criticalline, new estimates on the Gauss circle problem) dwell onexponential sum estimates of the type

‖∑l∼L

∑k∼K

akle(lx1 + klx2 + ω(k , l)x3)‖Lp(|x1|,|x2|<1,|x3|<(ηLK

12 )−1)

where

ω(k, l) =1

3((k + l)

32 − (k − l)3/2) = k1/2l + ck−3/2l3 + . . .

and η > 0 is a small parameter. Note that the points (l , kl , k12 l) sit

on the cone x = z2

y and the points (l , kl , ω(k , l)) sit on a slightperturbation of it.


Conjecture (Bourgain, D., 2014)

Assume F is supported in the δ-neighborhood of Co2. Then

‖F‖L4(R3) .ε δ−ε−( 1

2− 1

4)(

∑γ∈Θboxes

Co2 (δ)

‖PγF‖4L4(R3))1/4

Two failed lines of attack:• The approach we used to prove the similar result for surfaceswith nonzero curvature relies on parabolic rescaling. The analogousaffine maps that leave the cone invariant are the Lorentz maps L.The problem is that they do not stretch in the zero curvaturedirection, so square-like caps are not mapped to square-like caps.


• Assume F is supported in the δ-neighborhood of Co2. Combiningl2(L4) decoupling into planks with Holder gives l4(L4) decoupling

‖F‖L4(R3) .ε δ−ε− 1

2( 1

2− 1

4)(

∑θ∈Θplanks

Co2 (δ)

‖PθF‖4L4(R3))1/4.

Since θ is flat, the best we can say is (note the double exponent)

‖Fθ‖L4(R3) .ε δ−2× 1

2( 1

2− 1

4)(∑γ⊂θ‖PθF‖4

L4(R3))1/4.

Combining this leads to

‖F‖L4(R3) .ε δ−ε− 1

2( 1

2− 1

4)−( 1

2− 1

4)(

∑γ∈Θboxes

Co2 (δ)

‖PγF‖4L4(R3))1/4.

Unfortunately the exponent is now too large...Ciprian Demeter, IU Bloomington Refinements of Vinogradov’s Mean Value Theorem

The same kind of question can be asked for the parabola.Partition Γα(δ) of the δ-neighborhood NP1(δ) of the parabolainto boxes γ of length δα, α > 1/2

Conjecture (Small cap lp decoupling for the parabola)

Assume F : R2 → C has Fourier transform supported on NP1(δ).Then for each 2 ≤ p ≤ 2 + 2

α we have

‖F‖Lp(R2) .ε δ−α( 1

2− 1

p)−ε(

∑γ∈Γα(δ)

‖PγF‖pLp(R2))

1p .


Conjecture (Small cap lp decoupling for the parabola)

Assume F : R2 → C has Fourier transform supported on NP1(δ).Then for each 2 ≤ p ≤ 2 + 2

α we have

‖F‖Lp(R2) .ε δ−α( 1

2− 1

p)−ε(

∑γ∈Γα(δ)

‖PγF‖pLp(R2))

1p .

Known cases are α = 12 (Bourgain, D. 2014) and α = 1

(reformulation of the restriction conjecture for the parabola)

The approach (described for the cone) of combining thedecoupling into θ with flat decoupling for each θ fails.


Similar questions can be asked for the moment curvePartition into boxes γ of length δα, α > 1/3

Some motivation comes from the following conjecture

Conjecture (Heath-Brown)

For each 0 ≤ β ≤ n − 1 and s ≥ 1 we have∫[0,1]n−1

dx1 . . . dxn−1

∫ 1

Nβ

0|

N∑k=1

e(kx1 + k2x2 + . . .+ knxn)|2sdxn

.ε Nε(Ns−β + N2s− n(n+1)

2 ).




dx1 . . . dxn−1

∫ 1

Nβ

0|

N∑k=1

e(kx1 + k2x2 + . . .+ knxn)|2sdxn


2 ).

Known cases• The case n = 2 follows from Gauss sum estimates.• The case β = 0 is Vinogradov’s Mean Value Theorem• n = 3, β = 2 is due to Bombieri-Iwaniec, reproved by Bourgainusing decoupling.


We made progress on all these problems using a combination of• improved two-step decoupling• refined Kakeya-type estimates

Recall that all these problems involve two types of frequencyboxes. The larger boxes θ are the ones for which we have afavorable decoupling. The smaller boxes γ are the ones for whichwe want to prove a decoupling.


Wave packet decomposition of PθF

PθF =∑T∈T

wTWT

with wT ∈ C, WT (x) ≈ 1T (x)e(x · cθ) and T a collection ofpairwise disjoint boxes dual to θ. Let T be a tiling of Rn withlarger boxes τ dual to γ. Note that each T ∈ T belongs to aunique τ ∈ T . Using pigeonholing, we may assume(1) |wT | ∼ const for T ∈ T(2) N-statistics: For some N ≥ 1, each τ ∈ T contains either∼ N boxed T ∈ T or no such box.


Recall flat decoupling for arbitrary F (without N-statistics)

‖PθF‖Lp(Rn) . L2×( 12− 1

p)(

L∑i=1

‖PγiF‖pLp(Rn))

1p .

Theorem (Improved flat decoupling)

Assume F satisfies (1) and (2). Then

‖PθF‖Lp(Rn) . (L2

N)

12− 1

p (L∑

i=1

‖PγiF‖pLp(Rn))

1p .


Improved flat decoupling is just half of the improved two-stepdecoupling approach.The second (more involved) half is concerned with refining thedecoupling of F into boxex θ, in a way that is sensitive to thestatistics parameter N. To achieve this, we use refined Kakeyaestimates for various boxes such as plates, planks and tubes.

In the case of the problem for exponential sums, a key feature weexploit is a certain periodicity of the small boxes T , in addition tothe N-statistics.


New results (with Guth and Wang):

Theorem (D., Guth and Wang, Small cap lp decoupling for theparabola)

Assume F : R2 → C has Fourier transform supported on NP1(δ).Assume F is ”uniform.”Then for each 2 ≤ p ≤ 2 + 2

α we have

‖F‖Lp(R2) .ε δ−α( 1

2− 1

p)−ε(

∑γ∈Γα(δ)

‖PγF‖pLp(R2))

1p .


Theorem (Refined Kakeya (Guth, Solomon and Wang))

Let W ≥ 1. Let T be a collection of (δ, 1)-tubes inside [0, 1]2.Assume that for each ( 1

W , 1)-tube τ there are roughly N tubesT ∈ T lying inside τ and having the same orientation as τ .Assume r ≥ δ1−ε|T| for some ε > 0. Let Qr be a collection ofpairwise disjoint squares with side length δ that intersect at least rtubes T ∈ T. Then

|Qr | .ε δ−ε 1

W

|T|2

r2.

Figure: Two tubes τ , each with two smaller tubes T inside


We proved the following conjecture when n = 3 and 0 ≤ β ≤ 32 .



dx1 . . . dxn−1

∫ 1

Nβ

0|

N∑k=1

e(kx1 + k2x2 + . . .+ knxn)|2sdxn


2 ).

Corollary (Heath-Brown)

ζ(18/25 + it)� t3/40+ε


Conjecture (Reverse square function estimate for the cone)

If F is supported on the δ-neighborhood of Co2

‖F‖L4(R3) .ε δ−ε‖(

∑θ

|Pθ|2)1/2‖L4(R3)

Theorem (D., Guth and Wang)

Assume the reverse square function estimate for the cone. If F issupported on the δ-neighborhood of Co2

‖F‖L4(R3) .ε δ−ε−( 1

2− 1

4)(

∑γ∈Θboxes

Co2 (δ)

‖PγF‖4L4(R3))1/4


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