DECOUPLING

. We give a new proof of l 2 decoupling for the parabola inspired from eﬃcient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and diﬀerences between this new proof and eﬃcient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as l 2 L 2 decoupling, Bernstein’s inequality, and ball inﬂation come into play.


Introduction
For an interval J Ă r0, 1s and g : r0, 1s Ñ C, we define pE J gqpxq :" where epaq :" e 2πia .For an interval I, let P ℓ pIq be the partition of I into intervals of length ℓ.By writing P ℓ pIq, we are assuming that |I|{ℓ P N. We will also similarly define P ℓ pBq for squares B in R 2 .Next if B " Bpc, Rq is a square in R 2 centered at c of side length R, let We will always assume that our squares have sides parallel to the x and y-axis.We observe that 1 B ď 2 100 w B .For a function w, we define }f } L p pwq :" p ż R 2 |f pxq| p wpxq dxq 1{p .
For δ P N ´1 " tn ´1 : n P Nu, let Dpδq be the best constant such that }E r0,1s g} L 6 pBq ď Dpδqp ÿ JPP δ pr0,1sq for all g : r0, 1s Ñ C and all squares B in R 2 of side length δ ´2.Let D p pδq be the decoupling constant where the L 6 in ( 2) is replaced with L p .Since 1 B À w B , the triangle inequality combined with the Cauchy-Schwarz inequality shows that D p pδq À p δ ´1{2 for all 1 ď p ď 8.The l 2 decoupling theorem for the paraboloid proven by Bourgain and Demeter in [4] implies that for the parabola we have D p pδq À ε δ ´ε for 2 ď p ď 6 and this range of p is sharp.Decoupling-type inequalities were first studied by Wolff in [24].Following the proof of l 2 decoupling for the paraboloid by Bourgain and Demeter in [4], decoupling inequalities for various curves and surfaces have found many applications to analytic number theory (see for example [2,3,5,7,8,10,11,14,15,17]).Most notably is the proof of Vinogradov's mean value theorem by Bourgain-Demeter-Guth using decoupling for the moment curve t Þ Ñ pt, t 2 , . . ., t n q in [8].Wooley in [26] was also able to prove Vinogradov's mean value theorem using his nested efficient congruencing method.
This paper probes the connections between efficient congruencing and l 2 decoupling in the simplest case of the parabola.For a slightly different interpretation of the relation between efficient congruencing and decoupling for the cubic moment curve inspired from [16], see [12].See also [13] for an interpretation of [26] in the decoupling language which provides an alternative proof of decoupling for the moment curve in R d different from the proof in [8].
Our proof of l 2 decoupling for the parabola is inspired by the exposition of Wooley's efficient congruencing in Pierce's Bourbaki seminar exposition [21,Section 4].This proof will give the following result.In the context of discrete Fourier restriction, Theorem 1.1 implies that for all N sufficiently large and arbitrary sequence ta n u Ă l 2 , we have Finally we will let η be a Schwartz function such that η ě 1 Bp0,1q and supppp ηq Ă Bp0, 1q.For B " Bpc, Rq we also define η B pxq :" ηp x´c R q.In Section 2 we care about explicit constants and so we will use the explicit η constructed in Corollary 6.7.In particular, for this η, η B ď 10 2400 w B .For the remaining sections in this paper, we will ignore this constant.The most important facts about w B we will need are that w Bp0,Rq ˚wBp0,Rq À R 2 w Bp0,Rq and 1 Bp0,Rq ˚wBp0,Rq Á R 2 w Bp0,Rq from which we can derive all the other properties about weights we will use such as given a partition t∆u of B, ř ∆ w ∆ À w B and # pBpy,Rqq w Bp0,Rq pyq dy.
We refer the reader to [6,Section 4] and [19,Section 2.2] for some useful details and properties of the weights w B and η B .To keep the paper relatively self contained, we have also included proofs of these estimates in Section 6 with explicit constants.
1.2.Outline of proof of Theorem 1.1.Our argument is inspired by the discussion of efficient congruencing in [21,Section 4] which in turn is based off Heath-Brown's simplification [16] of Wooley's proof of the cubic case of Vinogradov's mean value theorem [25].Our first step, much like the first step in both efficient congruencing and decoupling for the parabola, is to bilinearize the problem.Throughout we will assume δ ´1 P N and ν P N ´1 X p0, 1{100q.
Fix arbitrary integers a, b ě 1. Suppose δ and ν were such that ν a δ ´1, ν b δ ´1 P N.This implies that δ ď minpν a , ν b q and the requirement that ν maxpa,bq δ ´1 P N is equivalent to having ν a δ ´1, ν b δ ´1 P N. For this δ and ν, let M a,b pδ, νq be the best constant such that ż for all squares B of side length δ ´2, g : r0, 1s Ñ C, and all intervals I P P ν a pr0, 1sq, I 1 P P ν b pr0, 1sq with dpI, I 1 q ě 3ν.We will say that such I and I 1 are 3ν-separated.Applying Hölder's inequality followed by the triangle inequality and the Cauchy-Schwarz inequality shows that M a,b pδ, νq is finite.This is not the only bilinear decoupling constant we can use (see ( 28) and (32) in Sections 4 and 5, respectively), but in this outline we will use (3) because it is closest to the one used in [21] and the one we will use in Section 2.
Our proof of Theorem 1.1 is broken into the following four lemmas.We state them below ignoring explicit constants for now.Lemma 1.2 (Parabolic rescaling).Let 0 ă δ ă σ ă 1 be such that σ, δ, δ{σ P N ´1.Let I be an arbitrary interval in r0, 1s of length σ.Then for every g : r0, 1s Ñ C and every square B of side length δ ´2.
Lemma 1.3 (Bilinear reduction).Suppose δ and ν were such that νδ ´1 P N. Then .Applying Lemma 1.4, we can move from M 1,1 to M 2,1 and then Lemma 1.5 allows us to move from M 2,1 to M 1,2 at the cost of a square root of Dpδ{νq.Applying Lemma 1.4 again moves us to M 2,4 .Repeating this we can eventually reach M 2 N ´1,2 N paying some Op1q power of ν ´1 and the value of the linear decoupling constants at various scales.This combined with Lemma 1.3 and the choice of ν " δ 1{2 N leads to the following result.
Lemma 1.6.Let N P N and suppose δ was such that δ ´1{2 N P N and 0 ă δ ă 100 ´2N .Then This then gives a recursion which shows that Dpδq À ε δ ´ε (see Section 2.3 for more details).
The proof of Lemma 1.2 is essentially a change of variables and applying the definition of the linear decoupling constant (some small technical issues arise because of the weight w B , see [19,Section 2.4]).The idea is that a cap on the paraboloid can be stretched to the whole paraboloid without changing any geometric properties.The bilinear reduction Lemma 1.3 follows from Hölder's inequality.The argument we use is from Tao's exposition on the Bourgain-Demeter-Guth proof of Vinogradov's mean value theorem [22].In general dimension, the multilinear reduction follows from a Bourgain-Guth argument (see [9] and [6,Section 8]).We note that if a and b are so large that ν a , ν b « δ then M a,b « Op1q and so the goal of the iteration is to efficiently move from small a and b to very large a and b.
Lemma 1.4 is the most technical of the four lemmas and is where we use a Fefferman-Cordoba argument in Section 2. We can still close the iteration with Lemma 1.4 replaced by M a,b À M b,b for 1 ď a ă b and M b,b À ν ´1{6 M 2b,b .Both these estimates come from the same proof of Lemma 1.4 and is how we approach the iteration in Sections 3 and 4 (see Lemmas 3.3 and 3.5 and their rigorous counterparts Lemmas 4.7 and 4.8).The proof of these lemmas is a consequence of l 2 L 2 decoupling and ball inflation.Finally, Lemma 1.5 is an application of Hölder's inequality and parabolic rescaling.[21,Section 4].The main object of iteration in [21,Section 4] is the following bilinear object We think of p as ν ´1, JpXq{X 3 as Dpδq, and p a`2b I 1 pX; a, bq{X 3 as M a,b pδ, νq 6 .We have the expressions JpXq{X 3 and p a`2b I 1 pX; a, bq{X 3 because heuristically assuming square root cancellation (ignoring X ε powers) we expect JpXq « X 3 and I 1 pX; a, bq « X 3 {p a`2b .This heuristic explains why In the definition of I 1 , the max ξ‰η pmod pq condition can be thought of as corresponding to the transversality condition that I 1 and I 2 are ν-separated intervals of length ν.The integral over p0, 1s 2 corresponds to an integral over B. Finally the expression | ÿ 1ďxďX x"ξ pmod p a q epα 1 x `α2 x 2 q|, can be thought of as corresponding to |E I g| for I an interval of length ν a and so the whole of I 1 pX; a, bq can be thought of as ş B |E I1 g| 2 |E I2 g| 4 where ℓpI 1 q " ν a and ℓpI 2 q " ν b with I 1 and I 2 are Opνq-separated.This will be our interpretation in Section 2.

Comparison with efficient congruencing as in
Interpreting the proof of Lemma 1.4 using the uncertainty principle, we reinterpret I 1 pX; a, bq as (ignoring weight functions) where I and I 1 are length ν a and ν b , respectively and are ν-separated.The uncertainty principle says that ( 4) is essentially equal to where I and I 1 are length ν and ν-separated.Note that when b " 1 this then is exactly equal to The interpretation given above is now similar to the A p object studied by Bourgain-Demeter in [6].
1.4.Overview.Theorem 1.1 will be proved in Section 2 via a Fefferman-Cordoba argument.This argument does not generalize to proving that D p pδq À ε δ ´ε except for p " 4, 6.However in Section 3, by the uncertainty principle we reinterpret a key lemma from Section 2 (Lemma 2.7) which allows us to generalize the argument in Section 2 so that it can work for all 2 ď p ď 6.We make this completely rigorous in Section 4 by defining a slightly different (but morally equivalent) bilinear decoupling constant.A basic version of the ball inflation inequality similar to that used in [6, Theorem 9.2] and [8,Theorem 6.6] makes an appearance.Finally in Section 5, we reinterpret the argument made in Section 4 and write an argument that is more like that given in [6].We create a 1-parameter family of bilinear constants which in some sense "interpolate" between the Bourgain-Demeter argument and our argument here.
The three arguments in Sections 2-5 are similar but will use slightly different bilinear decoupling constants.We will only mention explicit constants in Section 2. In Sections 4 and 5, for simplicity, we will only prove that Dpδq À ε δ ´ε.Because the structure of the iteration in Sections 4 and 5 is the same as that in Section 2, we obtain essentially the same quantitative bounds as in Theorem 1.1 when making explicit the bounds in Sections 4 and 5.
Finally, in Section 6, we include some discussion on the explicit constants for various estimates that we need for the proof of Theorem 1.1.
Acknowledgements.The author would like to thank Ciprian Demeter, Larry Guth, and his advisor Terence Tao for encouragement and many discussions on decoupling.The author would also like to thank Kevin Hughes and Trevor Wooley for a fruitful discussion on efficient congruencing at the Harmonic Analysis and Related Areas conference held by the Clay Math Institute at the University of Oxford in September 2017.The author is partially supported by NSF grants DGE-1144087 and DMS-1266164.Finally the author would like to thank the referee for detailed comments and suggestions.

Proof of Theorem 1.1
We recall the definition of the bilinear decoupling constant M a,b as in (3).The arguments in this section will rely strongly on the fact that the exponents in the definition of M a,b are 2 and 4, though we will only essentially use this in Lemma 2.7. Given For each J P P σ pr0, 1sq, combining Lemma 2.1 with Corollary 6.5 gives The trivial bound of Opν pa`2bq{6 δ ´1{2 q for M a,b pδ, νq is too weak for applications.We instead give another trivial bound that follows from parabolic rescaling.Proof.
Taking sixth roots then completes the proof of Lemma 2. Proof.Fix arbitrary I 1 and I 2 intervals of lengths ν 2b and ν b , respectively which are 3ν-separated.Hölder's inequality then gives Applying the definition of M b,2b and parabolic rescaling bounds the above by which completes the proof of Lemma 2.4.
Proof.Let tI i u ν ´1 i"1 " P ν pr0, 1sq.We have We first consider the diagonal terms.The triangle inequality followed by the Cauchy-Schwarz inequality gives that Parabolic rescaling and the Cauchy-Schwarz inequality bounds this by Therefore the first term in ( 5) is bounded above by Next we consider the off-diagonal terms.We have Hölder's inequality gives that and therefore from (3) (and using that νδ ´1 P N), the second term in ( 5) is bounded by Combining this with (6) and applying the definition of Dpδq then completes the proof of Lemma 2.5.

2.2.
A Fefferman-Cordoba argument.In the proof of Lemma 2.7 we need a version of M a,b with both sides being L 6 pw B q.The following lemma shows that these two constants are equivalent.
Lemma 2.6.Suppose δ and ν were such that for all squares B of side length δ ´2, g : r0, 1s Ñ C, and all 3ν-separated intervals I P P ν a pr0, 1sq and Proof.Fix arbitrary 3ν-separated intervals I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq.It suffices to assume that B is centered at the origin.Corollary 6.4 gives # pBpy,δ ´2qq w B pyq dy.
Unraveling the definition of L 6 # and applying the definition of M a,b gives that the above is pw Bpy,δ ´2 q q q 3 w B pyq dy where the second inequality is by Hölder's inequality and the third inequality is by Minkowski.Since B is centered at the origin, w B ˚wB ď 4 100 δ ´4w B (Lemma 6.2) and hence This then immediately implies that M 1 a,b pδ, νq ď 12 100{6 M a,b pδ, νq which completes the proof of Lemma 2.6.
We have the following key technical lemma of this paper.We encourage the reader to compare the argument with that of [21,Lemma 4.4].
Proof.It suffices to assume that B is centered at the origin with side length δ ´2.The integrality conditions on δ and ν imply that δ ď ν 2b and ν a δ ´1, ν b δ ´1 P N. Fix arbitrary intervals I 1 " rα, α `νa s P P ν a pr0, 1sq and I 2 " rβ, β `νb s P P ν b pr0, 1sq which are 3ν-separated.
Suppose J 1 , J 2 P P ν 2b prd, d `νa sq such that dpJ 1 , J 2 q ą 10ν 2b´1 .Expanding the integral in (9) for this pair of J 1 , J 2 gives that it is equal to ż where the expression inside the ep¨¨¨q is 6 qx 2 q.Interchanging the integrals in ξ and x shows that the integral in x is equal to the inverse Fourier transform of η 100B evaluated at Since the inverse Fourier transform of η 100B is supported in Bp0, δ 2 {100q, (10) is equal to 0 unless Since δ ď ν 2b and ξ i P r0, ν b s for i " 2, 3, 5, 6, (11) implies Since I 1 , I 2 are 3ν-separated, |d| ě 3ν.Recall that ξ 1 P J 1 , ξ 4 P J 2 and J 1 , J 2 are subsets of rd, d `νa s.Write ξ 1 " d `r and ξ 4 " d `s with r, s P r0, ν a s.Then Since dpJ 1 , J 2 q ą 10ν 2b´1 , |ξ 1 ´ξ4 | ą 10ν 2b´1 .Therefore the left hand side of ( 12) is ą 40ν 2b , a contradiction.Thus the integral in ( 9) is equal to 0 when dpJ 1 , J 2 q ą 10ν 2b´1 .The above analysis implies that ( 9) is Undoing the change of variables as in (8) gives that the above is equal to ÿ Observe that η 100B pT β xq ď 10 2400 w 100B pT β xq ď 10 2600 w 100B pxq ď 10 2800 w B pxq where the second inequality is an application of Lemma 6.1 and the last inequality is because w B pxq ´1w 100B pxq ď 10 200 .An application of the Cauchy-Schwarz inequality shows that ( 14) is Note that for each J 1 P P ν 2b pI 1 q, there are ď 10000ν ´1 intervals J 2 P P ν 2b pI 1 q such that dpJ 1 , J 2 q ď 10ν 2b´1 .Thus two applications of the Cauchy-Schwarz inequality bounds the above by Since there are ď 10000ν ´1 relevant J 2 for each J 1 , the above is where the last inequality is an application of Lemma 2.6.This completes the proof of Lemma 2.7.
Iterating Lemmas 2.4 and 2.7 repeatedly gives the following estimate.
Remark 2. A similar analysis as in ( 11)- (13) shows that if 1 ď a ă b and δ and ν were such that ν b δ ´1 P N, then M a,b pδ, νq À M b,b pδ, νq.Though we do not iterate this way in this section, it is enough to close the iteration with M a,b À M b,b for 1 ď a ă b, and M b,b À ν ´1{6 M 2b,b , and Lemma 2.4.We interpret the iteration and in particular Lemma 2.7 this way in Sections 3-5.
Choose N to be an integer such that Then by Corollary 2.10, for δ ´1{2 N P N with δ ă 100 ´2N , where in the last inequality we have used (16).Applying almost multiplicativity of the linear decoupling constant (similar to [19, Section 2.10] or the proof of Lemma 2.12 later) then shows that for all δ P N ´1, This then contradicts minimality of λ.Therefore λ " 0 and thus we have shown that Dpδq À ε δ ´ε for all δ P N ´1.
Note that Lemma 2.11 is only true for δ satisfying δ ´1{2 N P N and δ ă 100 ´2N .We now use almost multiplicativity to upgrade the result of Lemma 2.11 to all δ P N ´1.
and δ P t2 ´2N n u 8 n"7 " tδ n u 8 n"7 .Then for these δ, δ ´1{2 N P N and δ ă 100 If δ P pδ n`1 , δ n s for some n ě 7, then almost multiplicativity and Lemma 2.11 gives that where N is as in (19) and the second inequality we have used the trivial bound for Dpδ{δ n q.Combining both cases above then shows that if N is chosen as in (19), then for all δ P N ´1.Since we are no longer constrained by having N P N, we can increase N to be 3{ε and so we have that ε δ ´ε for all δ P N ´1.This completes the proof of Lemma 2.12.
Optimizing in ε then gives the proof of our main result.
Proof of Theorem 1.1.Choose A " plog 2 200qplog 1 δ q, η " log A ´log log A, and ε " 1 η log 200.Note that η exppηq " Ap1 ´log log A log A q ď A. Then from our choice of η, A, ε, Since η " log A ´log log A, we need to ensure that our choice of ε is such that 0 ă ε ă 1{100.Thus we need .
Note that for all x ą 0, log log x ă plog xq 1{2 and hence for all 0 ă δ ă e Thus we need 0 ă δ ă e and hence δ ă e ´200 200 .Therefore using ( 20) and ( 21), we have that for δ P p0, e ´200 200 q X N ´1, This completes the proof of Theorem 1.1.
3. An uncertainty principle interpretation of Lemma 2.7 We now give a different interpretation of Lemma 2.7, making use of the uncertainty principle.We will pretend all weight functions w B are indicator functions 1 B in this section and will make the argument rigorous in the next section.In this section, B will denote an arbitrary square of side length δ ´2.
The main point was of Lemma 2.7 was to show that if 1 ď a ď 2b, δ and ν such that ν 2b δ ´1 P N, then ż for arbitrary I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq such that dpI 1 , I 2 q Á ν.From Lemma 2.8, we only need (22) to be true for 1 ď a ď b.Our goal of this section is to prove (heuristically under the uncertainty principle) the following two statements: for arbitrary I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq such that dpI 1 , I 2 q Á ν.
for arbitrary I 1 , I 2 P P ν b pr0, 1sq such that dpI 1 , I 2 q Á ν.Replacing 4 with p ´2 then allows us to generalize to 2 ď p ă 6.
The particular instance of the uncertainty principle we will use is the following.Let I be an interval of length 1{R with center c.Fix an arbitrary R ˆR2 rectangle T oriented in the direction p´2c, 1q.Heuristically for x P T , pE I gqpxq behaves like a T,I e 2πiωT,I ¨x.Here the amplitude a T depends on g, T , and I and the phase ω T,I depends on T and I.In particular, |pE I gqpxq| is essentially constant on every R ˆR2 rectangle oriented in the direction p´2c, 1q.This also implies that if ∆ is a square of side length R, then |pE I gqpxq| is essentially constant on ∆ (with constant depending on ∆, I, g) and }E I g} L p # p∆q is essentially constant with the same constant independent of p.
Lemma 3.1 (Bernstein's inequality).Let I be an interval of length 1{R and ∆ a square of side length R. If 1 ď p ď q ă 8, then The reverse inequality in the above lemma is just an application of Hölder's inequality and so ignoring weight functions, }E I g} L q # p∆q " }E I g} L p # p∆q for any 1 ď p, q ď 8.In other words, }E I g} L p # p∆q is essentially constant independent of p. Therefore we can view Bernstein's inequality as one instance of the uncertainty principle.
Lemma 3.2 (l 2 L 2 decoupling).Let I be an interval of length ě 1{R such that R|I| P N and ∆ a square of side length R. Then Proof.See [6, Proposition 6.1] or [19, Lemma 2.2.21] for a rigorous proof.
The first inequality ( 23) is an immediate application of the uncertainty principle and l 2 L 2 decoupling.Lemma 3.3.Suppose 1 ď a ă b and δ and ν were such that ν b δ ´1 P N. Then ż for arbitrary I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq such that dpI 1 , I 2 q Á ν.In other words, M a,b pδ, νq À M b,b pδ, νq.
Proof.It suffices to show that for each ∆ 1 P P ν ´b pBq, we have ż Since I 2 is an interval of length ν b , |E I2 g| is essentially constant on ∆ 1 .Therefore the above reduces to showing ż which since a ă b and I 1 is of length ν a is just an application of l 2 L 2 decoupling.This completes the proof of Lemma 3.3.
Inequality ( 24) is a consequence of the following ball inflation lemma which is reminiscent of the ball inflation in the Bourgain-Demeter-Guth proof of Vinogradov's mean value theorem.The main point of this lemma is to increase the spatial scale so we can apply l 2 L 2 decoupling while keeping the frequency scales constant.
Lemma 3.4 (Ball inflation).Let b ě 1 be a positive integer.Suppose I 1 and I 2 are intervals of length ν b with dpI 1 , I 2 q Á ν.Then for any square ∆ 1 of side length ν ´2b , we have Proof.The uncertainty principle implies that |E I1 g| and |E I2 g| are essentially constant on ∆.Therefore we essentially have Avg Cover ∆ 1 by disjoint rectangles tT 1 u of size ν ´b ˆν´2b pointing in the direction p´2c I1 , 1q where c I1 is the center of I ! .Similarly form the collection of ν ´b ν´2b rectangles tT 2 u corresponding to I 2 .From the uncertainty principle, |E I1 g| " ř T1 |a T1 |1 T1 and |E I2 g| " ř T2 |a T2 |1 T2 for some constants |a Ti | depending on T i , g, and ∆ 1 .
Since I 1 and I 2 are Opνq-separated, for any two tubes T 1 , T 2 corresponding to and |∆ 1 | " ν ´4b , this completes the proof of Lemma 3.4.
We now prove inequality (24).Proof.This is an application of ball inflation, l 2 L 2 decoupling, Bernstein's inequality, and the uncertainty principle.Since ν 2b δ ´1 P N, ν b δ ´1 P N and δ ď ν 2b .Fix arbitrary I 1 , I 2 P P ν b pr0, 1sq.We have where the second inequality is because of Bernstein's inequality.From ball inflation we know that for each ∆ 1 P P ν ´2b pBq, Averaging the above over all ∆ 1 P P ν ´2b pBq shows that ( 25) is Since I 1 is of length ν b , l 2 L 2 decoupling gives that the above is Since |E J g| is essentially constant on ∆ 1 , the uncertainty principle gives that essentially we have Combining the above two centered equations then completes the proof of Lemma 3.5.
Remark 3. The proof of Lemma 3.5 is reminiscent of our proof of Lemma 2.7.The }E I2 g} L 8 p∆q can be thought as using the trivial bound for ξ i , i " 2, 3, 5, 6 to obtain (12).Then we apply some data about separation, much like in ball inflation here to get large amounts of cancelation.
Remark 4. After the submission of this manuscript, the author along with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich were able to interpret Wooley's nested efficient congruencing paper [26] in terms of decoupling which gave a new rather short proof of l 2 decoupling for the moment curve in R k [13].Restricting our paper to k " 2 gives a third proof of Lemma 3.5 that just uses Plancherel's theorem.
In [13] we use the Fourier supported in a neighborhood formulation of decoupling.
In what follows we give a heuristic sketch of the argument using the formulation of decoupling with an extension operator.See [13] or [23,Proposition 19] for a rigorous proof.One can also make the argument rigorous using the methods in [12].By affine invariance of the parabola, we may assume that I 2 " r´ν b {2, ν b {2s and I 1 " rd, d `νa s where d Á ν.From the uncertainty principle, since I 2 " r´ν b {2, ν b {2s, |pE I2 gqpxq| is essentially constant on any vertical ν ´b ˆν´2b rectangle.Partition B into vertical ν ´b ˆν´2b rectangles l.It suffices to prove that for each l, we have ż Since |E I2 g| is essentially constant on l and appears on both sides, it suffices to prove ż We may assume that l " r0, ν ´bs ˆr0, ν ´2b s.It is enough to prove that each fixed x P r0, ν ´bs, we have We claim that this follows from Plancherel's theorem.Observe that |pE I gqpx, yq| " ηxq.Let Pprd 2 , pd `νa q 2 sq be the partition of this interval into intervals rd 2 , pd `ν2b q 2 s, rpd `ν2b q 2 , pd `2ν 2b q 2 s, rpd `2ν 2b q 2 , pd `3ν 2b q 2 s, etc.Let ψ r0,ν ´2b s be a Schwartz function such that ψ r0,ν ´2b s ě 1 r0,ν ´2b s and suppp p ψ r0,ν ´2b s q Ă r´ν 2b {2, ν 2b {2s.Then by Plancherel's theorem, Since the |J| " 2dν 2b `Opν 2b`a q, the G x 1 J ˚p ψ r0,ν ´2b s have almost pairwise disjoint support, and so the above is (essentially) This proves (26) and hence proves Lemma 3.5.

4.
An alternate proof of Dpδq À ε δ ´ε The ball inflation lemma and our proof of Lemma 3.5 inspire us to define a new bilinear decoupling constant that can make our uncertainty principle heuristics from the previous section rigorous.
The left hand side of the definition of Dpδq in ( 2) is unweighted, however observe that Corollary 6.5 implies that }E r0,1s g} L 6 pwB q À Dpδqp ÿ for all g : r0, 1s Ñ C and squares B of side length δ ´2.
We will assume that δ ´1 P N and ν P N ´1 X p0, 1{100q.Let M a,b pδ, νq be the best constant such that Avg ∆PP ν ´maxpa,bq pBq # pwB q q 2 (28) for all squares B of side length δ ´2, g : r0, 1s Ñ C and all intervals I P P ν a pr0, 1sq, I 1 P P ν b pr0, 1sq with dpI, I 1 q ě ν.
Suppose a ą b (the proof when a ď b is similar).The uncertainty principle implies that Avg where the last " is because |E I1 g| is essentially constant on ∆.Therefore our bilinear constant M a,b is essentially the same as the bilinear constant M a,b we defined in (3).
where the last inequality we have used that ř ∆ w ∆ À w B (see for example Corollary 6.3).Finally applying (27) with parabolic rescaling then completes the proof of Lemma 4.3.
Proof.Let I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq.We have where the first and second inequalities are because of Hölder's inequality and the third inequality is an application of Hölder's inequality and the estimate ř ∆ w ∆ À w B .Applying parabolic rescaling and the definition of M b,a then completes the proof of Lemma 4.4.Lemma 4.5 (Bilinear reduction).Suppose δ and ν were such that νδ ´1 P N. Then Proof.The proof is essentially the same as that of Lemma 2.5 except when analyzing (7) in the off-diagonal terms we use where the second inequality we have used Bernstein's inequality.
4.2.Ball inflation.We now prove rigorously the ball inflation lemma we mentioned in the previous section.
Lemma 4.6 (Ball inflation).Let b ě 1 be a positive integer.Suppose I 1 and I 2 are ν-separated intervals of length ν b .Then for any square ∆ 1 of side length ν ´2b , we have Proof.Observe that if c " pc 1 , c 2 q, then pE I gqpx `cq " pE I g c qpxq where g c pξq " gpξqepξc 1 `ξ2 c 2 q.Therefore we may assume that ∆ 1 is centered at the origin.Fix intervals I 1 and I 2 intervals of length ν b which are ν-separated with centers c 1 and c 2 , respectively.
Cover ∆ 1 by a set T 1 of mutually parallel nonoverlapping rectangles T 1 of dimensions ν ´b ˆν´2b with longer side pointing in the direction of p´2c 1 , 1q (the normal direction of the piece of parabola above I 1 ).Note that any such ν ´b ˆν´2b rectangle outside 4∆ 1 cannot cover ∆ 1 itself.Thus we may assume that all rectangles in T 1 are contained in 4∆ 1 .Finally let T 1 pxq be the rectangle in T 1 containing x. Similarly define T 2 except this time we use I 2 .
For x P 4∆ 1 , define Given a ∆ P P ν ´b p∆ 1 q, if x P ∆, then ∆ Ă 2T i pxq.This implies that the center of ∆, c ∆ P 2T i pxq for x P ∆, and hence for all x P ∆, By how for some constants a Ti ě 0. Thus using (30) and that the T i are disjoint, the left hand side of (29) is bounded above by where in the last inequality we have used that since I 1 and I 2 are ν-separated, sine of the angle between T 1 and T 2 is Á ν and hence and Therefore (31) is # p4∆ 1 q .Thus we are done if we can prove that # pw ∆ 1 q , but this was exactly what was shown in [6, Eq. ( 29)] (and [19, Lemma 2.6.3] for the same inequality but with explicit constants).
Our choice of bilinear constant (28) makes the rigorous proofs of Lemmas 3.3 and 3.5 immediate consequences of ball inflation and l 2 L 2 decoupling.Proof.For arbitrary I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq which are ν-separated, it suffices to show that Avg But this is immediate from l 2 L 2 decoupling which completes the proof of Lemma 4.7.Proof.For arbitrary I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq which are ν-separated, it suffices to prove that Avg But this is immediate from ball inflation followed by l 2 L 2 decoupling which completes the proof of Lemma 4.8.

4.3.
The O ε pδ ´εq bound.We now prove that Dpδq À ε δ ´ε.The structure of the argument is essentially the same as that in Section 2.3.Repeatedly iterating Corollary 4.9 and following the same steps in how we derived Lemma 2.8 and Corollaries 2.9 and 2.10 gives the following result.Lemma 4.10.Let N be an integer chosen sufficiently large later and let δ be such that δ ´1{2 N P N and 0 ă δ ă 100 ´2N .Then Trivial bounds for Dpδq show that 1 À Dpδq À δ ´1{2 for all δ P N ´1.Let λ be the smallest real number such that Dpδq À ε δ ´λ´ε for all δ P N ´1.From the trivial bounds λ P r0, 1{2s.We claim λ " 0. Suppose λ ą 0.
Let N be a sufficiently large integer ě 8 3λ .This implies Lemma 4.10 then implies that for δ such that δ ´1{2 N P N and 0 ă δ ă 100 ´2N , we have where the last inequality we have applied our choice of N .By almost multiplicity we then have the same estimate for all δ P N ´1 (with a potentially larger constant depending on N ).But this then contradicts minimality of λ.Therefore λ " 0.

Unifying two styles of proof
We now attempt to unify the Bourgain-Demeter style of decoupling and the style of decoupling mentioned in the previous section.In view of Corollary 4.9, instead of having two integer parameters a and b we just have one integer parameter.
Let b be an integer ě 1 and choose s P r2, 3s any real number.Suppose δ P N for all squares B of side length δ ´2, g : r0, 1s Ñ C, and all intervals I, I 1 P P ν pr0, 1sq which are ν-separated.Note that left hand side of the definition of M p3q b pδ, νq is the same as A 6 pq, B r , qq 6 defined in [6] and from the uncertainty principle, M p2q 1 pδ, νq is morally the same as M 1,1 pδ, νq defined in (3) and M 1,1 pδ, νq defined in (28).The l 2 piece in the definition of M psq b pδ, νq is chosen so that we can make the most out of applying l 2 L 2 decoupling.
We will use M psq b as our bilinear constant in this section to show that Dpδq À ε δ ´ε.The bilinear constant M psq b obeys essentially the same lemmas as in the previous sections.Proof.Fix arbitrary I 1 , I 2 P P ν pr0, 1sq which are ν-separated.Moving up from L 2 # to L 6 # followed by Hölder's inequality in the average over ∆ bounds the left hand side of (32) by # pw∆q q 6 2 q 6´s 6 .
Using Minkowski to switch the l 2 and l 6 sum followed by ř ∆ w ∆ À w B shows that this is The proof of Lemma 4.6 shows that this is Since for J P F 1 the values of }E J g} L s # pw ∆ 1 q are comparable and similarly for J 1 P F 2 , the above is This completes the proof of Lemma 5.3.
Lemma 5.4 (cf.Corollary 4.9).Suppose δ and ν were such that ν 2b δ ´1 P N. Then for every ε ą 0, Proof.Let θ and ϕ be such that where the first inequality is from Hölder's inequality and the second inequality is from ball inflation.We now use how θ and ϕ are defined to return to a piece which we control by l 2 L 2 decoupling and a piece which we can control by parabolic rescaling.Hölder's inequality (as in the definition of θ and ϕ) gives that the average above is bounded by Avg Hölder's inequality in the sum over J and J 1 shows that this is ď Avg Almost multiplicativity gives that Dpδq À N,ε δ ´λp1´1 2 N q´ε for all δ P N ´1, contradicting the minimality of λ.

Discussion on explicit constants
A close inspection of the proof of Theorem 1.1 reveals that there are two sources of explicit constants, one from the various weight functions adapted to B and another from parabolic rescaling (Lemma 2.1).To keep the paper as self contained as possible, we expand upon where the various explicit constants come from.Some details will only be briefly sketched as they can be found with explicit constants in [ In this notation, the weight function w B defined in ( 1) is equal to w B,100 .We include the dependence on E to distinguish between absolute constants and the dependence on the decay rate of w B,E and later in Lemma 6.11 we will need to use two different E. We also let Dpδ, Eq be the same definition as Dpδq in (2) except w B on the right hand side is replaced with w B,E .First we have an easy observation in how w B,E interacts with shear matrices.Lemma 6.1.Let S " p 1 a 0 1 q with |a| ď 2. Then w Bp0,Rq,E pSxq ď 3 E w Bp0,Rq,E pxq.
Next, we have the following key property of w B,E .Lemma 6.2.Let E ě 10.
We also have Proof.We first prove (37).We would like to give an upper bound for the expression In the case of the first integral in (40), pR 1 {Rq|y| ě |x| ´|x ´pR 1 {Rqy| ě |x|{2 and hence ż In the case of the second integral in (40), ż This then proves (37).
To prove (38) it suffices to give a lower bound for which depends only on E. As before, rescaling x and a change of variables in y gives that it suffices to give a lower bound independent of x for ż Bp0,1q This shows (38) and completes the proof of Lemma 6.2.
We have the following immediate corollaries.Proof.It suffices to prove the case when B is centered at the origin.Since B is a disjoint partition of B, ř ∆PB 1 ∆ " 1 B .Convolve both sides by w Bp0,R 1 q,E .For the left hand side use (38) and for the right hand side use that 1 B ď 2 E w B,E and (37).
Remark 5.The only property we needed in the above proof is that ř ∆PB 1 ∆ ď C1 B for some absolute constant C. In particular, the same proof will work with finitely overlapping covers and when R{R 1 R N. Corollary 6.4.For 1 ď p ă 8 and E ě 10, # pBpy,Rqq w Bp0,Rq,E pyq dy.
Proof.Expanding the right hand side, we see the expression 2 E R 2 1 Bp0,Rq ˚wBp0,Rq,E and then we use (38).Corollary 6.5.Let I Ă r0, 1s and P be a disjoint partition of I. Suppose for some 2 ď p ă 8, we have for all g : r0, 1s Ñ C and all squares B of side length R. Then for each E ě 10, we have for all g : r0, 1s Ñ C and all squares B of side length R.
Proof.The hypothesis and Corollary 6.4 imply that L p pw Bpy,Rq,E q q p{2 w B,E pyq dy.
Corollary 6.7.For x " px 1 , x 2 q, let ηpxq :" ψpx 1 qψpx 2 q.Fix a square B " Bpc, Rq of side length R. The function η B pxq " ηp x´c R q satisfies η B ě 1 B , supppx η B q Ă Bp0, 1{Rq, and for any E ě 100, η B pxq ď 2 2E E 4E w B,E pxq.6.3.Explicit proof of Lemma 2.1.We now discuss how we arrived at the explicit constant 10 16000 in Lemma 2.1.The argument we present here is slightly simpler than the one for the last centered equation in [19,Page 58], but the argument is essentially the same.
Then we apply the definition of D global pδ{σq and reverse the change of variables which completes the proof of Lemma 6.9.
Having proven Lemma 6.9 we are now almost done, essentially we just need to apply f " η B E I g to the above lemma and use that D global pδq À E Dpδ, Eq (as mentioned in [4, Remark 5.2] and essentially follows as a corollary from [6, Theorem 5.1]).There are two small but fixable problems with this argument.The first is that η B E I g is has Fourier support in a region slightly larger than θ I,δ and so pη B E I gq θ I,δ is not necessarily equal to η B E I g and so we will instead apply Lemma 6.9 to f " η B E ra`δ,a`σ´δs g.The second is that the À E in the estimate D global pδq À E Dpδ, Eq is not made explicit.Remark 6.To avoid the use of any equivalence of decoupling constants, one can instead just use Lemma 6.9 and suitably modify the Lemmas 2.2-2.7.This is the approach taken in [12,13] and in Tao's 247B notes on decoupling [23] (whose proof of parabola decoupling is based off the argument in this paper).We don't take this point of view here since it somewhat obscures the connection between efficient congruencing and decoupling, in particular when comparing the proof of Lemma 2.7 with [21,Lemma 4.4].Some equivalences between various decoupling constants were made quantitative by the author in [19,Proposition 2.3.11] and we will use this result here as a black box (the proof is quite similar to that of [6,Theorem 5.1]).To state the relevant part of [19,Proposition 2.3.11]used here, we define two more decoupling constants.Definition 6.10.For J " rn J δ, pn J `1qδs P P δ pr0, 1sq, let θ 1 J,δ :" tps, L J psq `tq : n J δ ď s ď pn J `1qδ, |t| ď 5δ 2 u where L J psq :" p2n J `1qδs ´nJ pn J `1qδ 2 and 0 ď n J ď δ ´1 ´1.Here θ 1 J,δ is a parallelogram that has height 10δ 2 and has base parallel to the straight line connecting pn J δ, n 2 J δ 2 q and ppn J `1qδ, pn J `1q 2 δ 2 q.

1 .
which rederives (up to constants) the upper bound obtained by Bourgain in [1, Proposition 2.36] but without resorting to use of a divisor bound.It is an open problem whether the exppOp log N log log N qq can be improved.1.More notation and weight functions.We define }f } L p # pBq :" p 1 |B| ż B |f pxq| p dxq 1{p , }f } L p # pwB q :" p 1 |B| ż |f | p w B q 1{p , and given a collection C of squares, we let Avg ∆PC

´1
and ν P N ´1 X p0, 1{100q were such that ν b δ ´1 P N. Let M psq b pδ, νq be the best constant such that Avg ∆PP ν ´b pBq p ÿ JPP ν b pIq

Corollary 6 . 3 .
Let B be a square of side length R and let B be a disjoint partition of B into squares ∆ with side length R 1 ă R. Then for E ě 10, ÿ ∆PB w ∆,E ď 16 E w B,E .
4.1.Some basic properties.We now have the weighted rigorous versions of Lemmas 3.1 and 3.2.Note that we will only need the L 8 version of Lemma 3.1.Lemma 4.2 (l 2 L 2 decoupling).Let I be an interval of length ě 1{R such that R|I| P N and ∆ a square of side length R. Then }E I g} L 2 pw∆q À p ÿ Lemma 4.1 (Bernstein's inequality).Let I be an interval of length 1{R and ∆ a square of side length R. Then}E I g} L 8 p∆q À }E I g} L p # pw∆q .JPP 1{R pIq }E J g} 2 L 2 pw∆q q 1{2 .We now run through the substitutes of Lemmas 2.3-2.5.Lemma 4.3.Suppose δ and ν were such that ν a δ ´1, ν b δ ´1 P N. ThenM a,b pδ, νq À Dp δ ν a q 1{3 Dp δ ν b q 2{3 .Proof.Let I 1 P P ν a pr0, 1sq and I 2 P P ν b pr0, 1sq.Hölder's inequality gives that ∆PP ν ´maxpa,bq pBq L s ď }f } θ L 2 }f } 1´θ L 6 and }f } L 6´s ď }f } ϕ L 2 }f } 1´ϕ L 6Fix arbitrary I 1 , I 2 P P ν pr0, 1sq which are ν-separated.We have 19, Sections 2.2-2.4].The argument we present here for the explicit argument in Lemma 2.1 is very slightly simpler and a bit different from that in [19, Section 2.4] but follows the same general philosophy.We claim no optimality in any explicit constant.6.1.Polynomial decaying weights w B,E .For x P R 2 and B a square centered at c P R 2 of side length R, let Schwartz function η B such that η B ě 1 B and supppx η B q Ă Bp0, 1{Rq.It is easy to justify existence of such a function, but we desire explicit quantitative estimates.A different Schwartz function was constructed in [19, Section 2.2.2] but the construction we provide here is slightly simpler in exposition.Lemma 6.6.Fix E ě 100.There exists a Schwartz function ψ on R such that ψ ě 1 r´1{2,1{2s , suppp p ψq Ă r´1{2, 1{2s and 6.2.Schwartz weight η B .Given B " Bpc, Rq, in this section we explicitly construct a 2 j 2j |x| j .and hence applying this bound to j " rE{2s gives For |x| ě 1, then p1 `|x|q E |ψpxq| ď 2 E E 2E and for |x| ď 1, p1 `|x|q E |ψpxq| ď 2 ¨2E .