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The following situation is very common in modern harmonic analysis: one has a large scale parameter (sometimes written as in the literature for some small scale parameter , or as for some large radius ), which ranges over some unbounded subset of (e.g. all sufficiently large real numbers , or all powers of two), and one has some positive quantity depending on that is known to be of *polynomial size* in the sense that

for all in the range and some constant , and one wishes to obtain a *subpolynomial upper bound* for , by which we mean an upper bound of the form

for all and all in the range, where can depend on but is independent of . In many applications, this bound is nearly tight in the sense that one can easily establish a matching lower bound

in which case the property of having a subpolynomial upper bound is equivalent to that of being *subpolynomial size* in the sense that

for all and all in the range. It would naturally be of interest to tighten these bounds further, for instance to show that is polylogarithmic or even bounded in size, but a subpolynomial bound is already sufficient for many applications.

Let us give some illustrative examples of this type of problem:

Example 1 (Kakeya conjecture)Here ranges over all of . Let be a fixed dimension. For each , we pick a maximal -separated set of directions . We let be the smallest constant for which one has the Kakeya inequalitywhere is a -tube oriented in the direction . The Kakeya maximal function conjecture is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

Example 2 (Restriction conjecture for the sphere)Here ranges over all of . Let be a fixed dimension. We let be the smallest constant for which one has the restriction inequalityfor all bounded measurable functions on the unit sphere equipped with surface measure , where is the ball of radius centred at the origin. The restriction conjecture of Stein for the sphere is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

Example 3 (Multilinear Kakeya inequality)Again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which aretransversein the sense that there is a uniform lower bound for the wedge product of directions for (equivalently, there is no hyperplane through the origin that intersects all of the ). For each , we let be the smallest constant for which one has the multilinear Kakeya inequalitywhere for each , is a collection of infinite tubes in of radius oriented in a direction in , which are separated in the sense that for any two tubes in , either the directions of differ by an angle of at least , or are disjoint; and is our notation for the geometric mean

The multilinear Kakeya inequality of Bennett, Carbery, and myself establishes that is of subpolynomial size; a later argument of Guth improves this further by showing that is bounded (and in fact comparable to ).

Example 4 (Multilinear restriction theorem)Once again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which aretransverseas in the previous example. For each , we let be the smallest constant for which one has the multilinear restriction inequalityfor all bounded measurable functions on for . Then the multilinear restriction theorem of Bennett, Carbery, and myself establishes that is of subpolynomial size; it is known to be bounded for (as can be easily verified from Plancherel’s theorem), but it remains open whether it is bounded for any .

Example 5 (Decoupling for the paraboloid)now ranges over the square numbers. Let , and subdivide the unit cube into cubes of sidelength . For any , define the extension operatorsand

for and . We also introduce the weight function

For any , let be the smallest constant for which one has the decoupling inequality

The decoupling theorem of Bourgain and Demeter asserts that is of subpolynomial size for all in the optimal range .

Example 6 (Decoupling for the moment curve)now ranges over the natural numbers. Let , and subdivide into intervals of length . For any , define the extension operatorsand more generally

for . For any , let be the smallest constant for which one has the decoupling inequality

It was shown by Bourgain, Demeter, and Guth that is of subpolynomial size for all in the optimal range , which among other things implies the Vinogradov main conjecture (as discussed in this previous post).

It is convenient to use asymptotic notation to express these estimates. We write , , or to denote the inequality for some constant independent of the scale parameter , and write for . We write to denote a bound of the form where as along the given range of . We then write for , and for . Then the statement that is of polynomial size can be written as

while the statement that has a subpolynomial upper bound can be written as

and similarly the statement that is of subpolynomial size is simply

Many modern approaches to bounding quantities like in harmonic analysis rely on some sort of *induction on scales* approach in which is bounded using quantities such as for some exponents . For instance, suppose one is somehow able to establish the inequality

for all , and suppose that is also known to be of polynomial size. Then this implies that has a subpolynomial upper bound. Indeed, one can iterate this inequality to show that

for any fixed ; using the polynomial size hypothesis one thus has

for some constant independent of . As can be arbitrarily large, we conclude that for any , and hence is of subpolynomial size. (This sort of iteration is used for instance in my paper with Bennett and Carbery to derive the multilinear restriction theorem from the multilinear Kakeya theorem.)

Exercise 7If is of polynomial size, and obeys the inequalityfor any fixed , where the implied constant in the notation is independent of , show that has a subpolynomial upper bound. This type of inequality is used to equate various linear estimates in harmonic analysis with their multilinear counterparts; see for instance this paper of myself, Vargas, and Vega for an early example of this method.

In more recent years, more sophisticated induction on scales arguments have emerged in which one or more auxiliary quantities besides also come into play. Here is one example, this time being an abstraction of a short proof of the multilinear Kakeya inequality due to Guth. Let be the quantity in Example 3. We define similarly to for any , except that we now also require that the diameter of each set is at most . One can then observe the following estimates:

- (Triangle inequality) For any , we have
- (Multiplicativity) For any , one has
- (Loomis-Whitney inequality) We have

These inequalities now imply that has a subpolynomial upper bound, as we now demonstrate. Let be a large natural number (independent of ) to be chosen later. From many iterations of (6) we have

and hence by (7) (with replaced by ) and (5)

where the implied constant in the exponent does not depend on . As can be arbitrarily large, the claim follows. We remark that a nearly identical scheme lets one deduce decoupling estimates for the three-dimensional cone from that of the two-dimensional paraboloid; see the final section of this paper of Bourgain and Demeter.

Now we give a slightly more sophisticated example, abstracted from the proof of decoupling of the paraboloid by Bourgain and Demeter, as described in this study guide after specialising the dimension to and the exponent to the endpoint (the argument is also more or less summarised in this previous post). (In the cited papers, the argument was phrased only for the non-endpoint case , but it has been observed independently by many experts that the argument extends with only minor modifications to the endpoint .) Here we have a quantity that we wish to show is of subpolynomial size. For any and , one can define an auxiliary quantity . The precise definitions of and are given in the study guide (where they are called and respectively, setting and ) but will not be of importance to us for this discussion. Suffice to say that the following estimates are known:

- (Crude upper bound for ) is of polynomial size: .
- (Bilinear reduction, using parabolic rescaling) For any , one has
- (Crude upper bound for ) For any one has
- (Application of multilinear Kakeya and decoupling) If are sufficiently small (e.g. both less than ), then

In all of these bounds the implied constant exponents such as or are independent of and , although the implied constants in the notation can depend on both and . Here we gloss over an annoying technicality in that quantities such as , , or might not be an integer (and might not divide evenly into ), which is needed for the application to decoupling theorems; this can be resolved by restricting the scales involved to powers of two and restricting the values of to certain rational values, which introduces some complications to the later arguments below which we shall simply ignore as they do not significantly affect the numerology.

It turns out that these estimates imply that is of subpolynomial size. We give the argument as follows. As is known to be of polynomial size, we have some for which we have the bound

for all . We can pick to be the minimal exponent for which this bound is attained: thus

We will call this the *upper exponent* of . We need to show that . We assume for contradiction that . Let be a sufficiently small quantity depending on to be chosen later. From (10) we then have

for any sufficiently small . A routine iteration then gives

for any that is independent of , if is sufficiently small depending on . A key point here is that the implied constant in the exponent is uniform in (the constant comes from summing a convergent geometric series). We now use the crude bound (9) followed by (11) and conclude that

Applying (8) we then have

If we choose sufficiently large depending on (which was assumed to be positive), then the negative term will dominate the term. If we then pick sufficiently small depending on , then finally sufficiently small depending on all previous quantities, we will obtain for some strictly less than , contradicting the definition of . Thus cannot be positive, and hence has a subpolynomial upper bound as required.

Exercise 8Show that one still obtains a subpolynomial upper bound if the estimate (10) is replaced withfor some constant , so long as we also improve (9) to

(This variant of the argument lets one handle the non-endpoint cases of the decoupling theorem for the paraboloid.)

To establish decoupling estimates for the moment curve, restricting to the endpoint case for sake of discussion, an even more sophisticated induction on scales argument was deployed by Bourgain, Demeter, and Guth. The proof is discussed in this previous blog post, but let us just describe an abstract version of the induction on scales argument. To bound the quantity , some auxiliary quantities are introduced for various exponents and and , with the following bounds:

- (Crude upper bound for ) is of polynomial size: .
- (Multilinear reduction, using non-isotropic rescaling) For any and , one has
- (Crude upper bound for ) For any and one has
- (Hölder) For and one has
- (Rescaled decoupling hypothesis) For , one has
- (Lower dimensional decoupling) If and , then
- (Multilinear Kakeya) If and , then

It is now substantially less obvious that these estimates can be combined to demonstrate that is of subpolynomial size; nevertheless this can be done. A somewhat complicated arrangement of the argument (involving some rather unmotivated choices of expressions to induct over) appears in my previous blog post; I give an alternate proof later in this post.

These examples indicate a general strategy to establish that some quantity is of subpolynomial size, by

- (i) Introducing some family of related auxiliary quantities, often parameterised by several further parameters;
- (ii) establishing as many bounds between these quantities and the original quantity as possible; and then
- (iii) appealing to some sort of “induction on scales” to conclude.

The first two steps (i), (ii) depend very much on the harmonic analysis nature of the quantities and the related auxiliary quantities, and the estimates in (ii) will typically be proven from various harmonic analysis inputs such as Hölder’s inequality, rescaling arguments, decoupling estimates, or Kakeya type estimates. The final step (iii) requires no knowledge of where these quantities come from in harmonic analysis, but the iterations involved can become extremely complicated.

In this post I would like to observe that one can clean up and made more systematic this final step (iii) by passing to upper exponents (12) to eliminate the role of the parameter (and also “tropicalising” all the estimates), and then taking similar limit superiors to eliminate some other less important parameters, until one is left with a simple linear programming problem (which, among other things, could be amenable to computer-assisted proving techniques). This method is analogous to that of passing to a simpler asymptotic limit object in many other areas of mathematics (for instance using the Furstenberg correspondence principle to pass from a combinatorial problem to an ergodic theory problem, as discussed in this previous post). We use the limit superior exclusively in this post, but many of the arguments here would also apply with one of the other generalised limit functionals discussed in this previous post, such as ultrafilter limits.

For instance, if is the upper exponent of a quantity of polynomial size obeying (4), then a comparison of the upper exponent of both sides of (4) one arrives at the scalar inequality

from which it is immediate that , giving the required subpolynomial upper bound. Notice how the passage to upper exponents converts the estimate to a simpler inequality .

Exercise 9Repeat Exercise 7 using this method.

Similarly, given the quantities obeying the axioms (5), (6), (7), and assuming that is of polynomial size (which is easily verified for the application at hand), we see that for any real numbers , the quantity is also of polynomial size and hence has some upper exponent ; meanwhile itself has some upper exponent . By reparameterising we have the homogeneity

for any . Also, comparing the upper exponents of both sides of the axioms (5), (6), (7) we arrive at the inequalities

For any natural number , the third inequality combined with homogeneity gives , which when combined with the second inequality gives , which on combination with the first estimate gives . Sending to infinity we obtain as required.

Now suppose that , obey the axioms (8), (9), (10). For any fixed , the quantity is of polynomial size (thanks to (9) and the polynomial size of ), and hence has some upper exponent ; similarly has some upper exponent . (Actually, strictly speaking our axioms only give an upper bound on so we have to temporarily admit the possibility that , though this will soon be eliminated anyway.) Taking upper exponents of all the axioms we then conclude that

for all and .

Assume for contradiction that , then , and so the statement (20) simplifies to

At this point we can eliminate the role of and simplify the system by taking a second limit superior. If we write

then on taking limit superiors of the previous inequalities we conclude that

for all ; in particular . We take advantage of this by taking a further limit superior (or “upper derivative”) in the limit to eliminate the role of and simplify the system further. If we define

so that is the best constant for which as , then is finite, and by inserting this “Taylor expansion” into the right-hand side of (21) and conclude that

This leads to a contradiction when , and hence as desired.

Exercise 10Redo Exercise 8 using this method.

The same strategy now clarifies how to proceed with the more complicated system of quantities obeying the axioms (13)–(19) with of polynomial size. Let be the exponent of . From (14) we see that for fixed , each is also of polynomial size (at least in upper bound) and so has some exponent (which for now we can permit to be ). Taking upper exponents of all the various axioms we can now eliminate and arrive at the simpler axioms

for all , , and , with the lower dimensional decoupling inequality

for and , and the multilinear Kakeya inequality

for and .

As before, if we assume for sake of contradiction that then the first inequality simplifies to

We can then again eliminate the role of by taking a second limit superior as , introducing

and thus getting the simplified axiom system

for and , and

for and .

In view of the latter two estimates it is natural to restrict attention to the quantities for . By the axioms (22), these quantities are of the form . We can then eliminate the role of by taking another limit superior

The axioms now simplify to

It turns out that the inequality (27) is strongest when , thus

From the last two inequalities (28), (29) we see that a special role is likely to be played by the exponents

for and

for . From the convexity (25) and a brief calculation we have

for , hence from (28) we have

Similarly, from (25) and a brief calculation we have

for ; the same bound holds for if we drop the term with the factor, thanks to (24). Thus from (29) we have

for , again with the understanding that we omit the first term on the right-hand side when . Finally, (26) gives

Let us write out the system of equations we have obtained in full:

We can then eliminate the variables one by one. Inserting (33) into (32) we obtain

which simplifies to

Inserting this into (34) gives

which when combined with (35) gives

which simplifies to

Iterating this we get

for all and

for all . In particular

which on insertion into (36), (37) gives

which is absurd if . Thus and so must be of subpolynomial growth.

Remark 11(This observation is essentially due to Heath-Brown.) If we let denote the column vector with entries (arranged in whatever order one pleases), then the above system of inequalities (32)–(36) (using (37) to handle the appearance of in (36)) readsfor some explicit square matrix with non-negative coefficients, where the inequality denotes pointwise domination, and is an explicit vector with non-positive coefficients that reflects the effect of (37). It is possible to show (using (24), (26)) that all the coefficients of are negative (assuming the counterfactual situation of course). Then we can iterate this to obtain

for any natural number . This would lead to an immediate contradiction if the Perron-Frobenius eigenvalue of exceeds because would now grow exponentially; this is typically the situation for “non-endpoint” applications such as proving decoupling inequalities away from the endpoint. In the endpoint situation discussed above, the Perron-Frobenius eigenvalue is , with having a non-trivial projection to this eigenspace, so the sum now grows at least linearly, which still gives the required contradiction for any . So it is important to gather “enough” inequalities so that the relevant matrix has a Perron-Frobenius eigenvalue greater than or equal to (and in the latter case one needs non-trivial injection of an induction hypothesis into an eigenspace corresponding to an eigenvalue ). More specifically, if is the spectral radius of and is a left Perron-Frobenius eigenvector, that is to say a non-negative vector, not identically zero, such that , then by taking inner products of (38) with we obtain

If this leads to a contradiction since is negative and is non-positive. When one still gets a contradiction as long as is strictly negative.

Remark 12(This calculation is essentially due to Guo and Zorin-Kranich.) Here is a concrete application of the Perron-Frobenius strategy outlined above to the system of inequalities (32)–(37). Consider the weighted sumI had secretly calculated the weights , as coming from the left Perron-Frobenius eigenvector of the matrix described in the previous remark, but for this calculation the precise provenance of the weights is not relevant. Applying the inequalities (31), (30) we see that is bounded by

(with the convention that the term is absent); this simplifies after some calculation to the bound

and this and (37) then leads to the required contradiction.

Exercise 13

- (i) Extend the above analysis to also cover the non-endpoint case . (One will need to establish the claim for .)
- (ii) Modify the argument to deal with the remaining cases by dropping some of the steps.

In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely

Theorem 1 (Two-dimensional Vinogradov main conjecture)One hasas .

This particular case of the main conjecture has a classical proof using some elementary number theory. Indeed, the left-hand side can be viewed as the number of solutions to the system of equations

with . These two equations can combine (using the algebraic identity applied to ) to imply the further equation

which, when combined with the divisor bound, shows that each is associated to choices of excluding diagonal cases when two of the collide, and this easily yields Theorem 1. However, the Bourgain-Demeter-Guth argument (which, in the two dimensional case, is essentially contained in a previous paper of Bourgain and Demeter) does not require the divisor bound, and extends for instance to the the more general case where ranges in a -separated set of reals between to .

In this special case, the Bourgain-Demeter argument simplifies, as the lower dimensional inductive hypothesis becomes a simple almost orthogonality claim, and the multilinear Kakeya estimate needed is also easy (collapsing to just Fubini’s theorem). Also one can work entirely in the context of the Vinogradov main conjecture, and not turn to the increased generality of decoupling inequalities (though this additional generality is convenient in higher dimensions). As such, I am presenting this special case as an introduction to the Bourgain-Demeter-Guth machinery.

We now give the specialisation of the Bourgain-Demeter argument to Theorem 1. It will suffice to establish the bound

for all , (where we keep fixed and send to infinity), as the bound then follows by combining the above bound with the trivial bound . Accordingly, for any and , we let denote the claim that

as . Clearly, for any fixed , holds for some large , and it will suffice to establish

Proposition 2Let , and let be such that holds. Then there exists (with depending continuously on ) such that holds.

Indeed, this proposition shows that for , the infimum of the for which holds is zero.

We prove the proposition below the fold, using a simplified form of the methods discussed in the previous blog post. To simplify the exposition we will be a bit cavalier with the uncertainty principle, for instance by essentially ignoring the tails of rapidly decreasing functions.

Given any finite collection of elements in some Banach space , the triangle inequality tells us that

However, when the all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. For instance, if is a Hilbert space and the are mutually orthogonal, we have the Pythagorean theorem

For sake of comparison, from the triangle inequality and Cauchy-Schwarz one has the general inequality

for any finite collection in any Banach space , where denotes the cardinality of . Thus orthogonality in a Hilbert space yields “square root cancellation”, saving a factor of or so over the trivial bound coming from the triangle inequality.

More generally, let us somewhat informally say that a collection exhibits *decoupling in * if one has the Pythagorean-like inequality

for any , thus one obtains almost the full square root cancellation in the norm. The theory of *almost orthogonality* can then be viewed as the theory of decoupling in Hilbert spaces such as . In spaces for one usually does not expect this sort of decoupling; for instance, if the are disjointly supported one has

and the right-hand side can be much larger than when . At the opposite extreme, one usually does not expect to get decoupling in , since one could conceivably align the to all attain a maximum magnitude at the same location with the same phase, at which point the triangle inequality in becomes sharp.

However, in some cases one can get decoupling for certain . For instance, suppose we are in , and that are *bi-orthogonal* in the sense that the products for are pairwise orthogonal in . Then we have

giving decoupling in . (Similarly if each of the is orthogonal to all but of the other .) A similar argument also gives decoupling when one has tri-orthogonality (with the mostly orthogonal to each other), and so forth. As a slight variant, Khintchine’s inequality also indicates that decoupling should occur for any fixed if one multiplies each of the by an independent random sign .

In recent years, Bourgain and Demeter have been establishing *decoupling theorems* in spaces for various key exponents of , in the “restriction theory” setting in which the are Fourier transforms of measures supported on different portions of a given surface or curve; this builds upon the earlier decoupling theorems of Wolff. In a recent paper with Guth, they established the following decoupling theorem for the curve parameterised by the polynomial curve

For any ball in , let denote the weight

which should be viewed as a smoothed out version of the indicator function of . In particular, the space can be viewed as a smoothed out version of the space . For future reference we observe a fundamental self-similarity of the curve : any arc in this curve, with a compact interval, is affinely equivalent to the standard arc .

Theorem 1 (Decoupling theorem)Let . Subdivide the unit interval into equal subintervals of length , and for each such , let be the Fourier transformof a finite Borel measure on the arc , where . Then the exhibit decoupling in for any ball of radius .

Orthogonality gives the case of this theorem. The bi-orthogonality type arguments sketched earlier only give decoupling in up to the range ; the point here is that we can now get a much larger value of . The case of this theorem was previously established by Bourgain and Demeter (who obtained in fact an analogous theorem for any curved hypersurface). The exponent (and the radius ) is best possible, as can be seen by the following basic example. If

where is a bump function adapted to , then standard Fourier-analytic computations show that will be comparable to on a rectangular box of dimensions (and thus volume ) centred at the origin, and exhibit decay away from this box, with comparable to

On the other hand, is comparable to on a ball of radius comparable to centred at the origin, so is , which is just barely consistent with decoupling. This calculation shows that decoupling will fail if is replaced by any larger exponent, and also if the radius of the ball is reduced to be significantly smaller than .

This theorem has the following consequence of importance in analytic number theory:

Corollary 2 (Vinogradov main conjecture)Let be integers, and let . Then

*Proof:* By the Hölder inequality (and the trivial bound of for the exponential sum), it suffices to treat the critical case , that is to say to show that

We can rescale this as

As the integrand is periodic along the lattice , this is equivalent to

The left-hand side may be bounded by , where and . Since

the claim now follows from the decoupling theorem and a brief calculation.

Using the Plancherel formula, one may equivalently (when is an integer) write the Vinogradov main conjecture in terms of solutions to the system of equations

but we will not use this formulation here.

A history of the Vinogradov main conjecture may be found in this survey of Wooley; prior to the Bourgain-Demeter-Guth theorem, the conjecture was solved completely for , or for and either below or above , with the bulk of recent progress coming from the *efficient congruencing* technique of Wooley. It has numerous applications to exponential sums, Waring’s problem, and the zeta function; to give just one application, the main conjecture implies the predicted asymptotic for the number of ways to express a large number as the sum of fifth powers (the previous best result required fifth powers). The Bourgain-Demeter-Guth approach to the Vinogradov main conjecture, based on decoupling, is ostensibly very different from the efficient congruencing technique, which relies heavily on the arithmetic structure of the program, but it appears (as I have been told from second-hand sources) that the two methods are actually closely related, with the former being a sort of “Archimedean” version of the latter (with the intervals in the decoupling theorem being analogous to congruence classes in the efficient congruencing method); hopefully there will be some future work making this connection more precise. One advantage of the decoupling approach is that it generalises to non-arithmetic settings in which the set that is drawn from is replaced by some other similarly separated set of real numbers. (A random thought – could this allow the Vinogradov-Korobov bounds on the zeta function to extend to Beurling zeta functions?)

Below the fold we sketch the Bourgain-Demeter-Guth argument proving Theorem 1.

I thank Jean Bourgain and Andrew Granville for helpful discussions.

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