Selected problems on exceptional sets

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He is also known for the theory of Carleson measures. In the theory of dynamical systems , Carleson has worked in complex dynamics. In addition to publishing some landmark papers, Carleson has also published two books: First, an influential book on potential theory , "Selected Problems on Exceptional Sets" Van Nostrand, , and second a book on the iteration of analytic functions , Complex Dynamics Springer, , in collaboration with T.

He was awarded the Wolf Prize in Mathematics in , the Lomonosov Gold Medal in , the Sylvester Medal in , and the Abel Prize in for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems. He is a member of the Norwegian Academy of Science and Letters. From Wikipedia, the free encyclopedia. This biography of a living person needs additional citations for verification.

Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately , especially if potentially libelous or harmful. Notices of the American Mathematical Society. Retrieved Norwegian Academy of Science and Letters. Archived from the original on 10 November Retrieved 7 October Laureates of the Wolf Prize in Mathematics. Stein Mumford Abel Prize laureates. Using a Carleman estimate first introduced by Escauriaza, Seregin, and Sverak , as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time , establishing a lower bound for the vorticity on the spatial annulus.

By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the norm of the velocity ; crucially, this lower bound is uniform in. If is very large triple exponential in! The chain of causality is summarised in the following image:. It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the upper bound costs the third.

This note gives two proofs of a general eigenvector identity observed recently by Denton, Parke and Zhang in the course of some quantum mechanical calculations. The identity is as follows:. Theorem 1 Let be an Hermitian matrix, with eigenvalues. Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of. Once one is aware of the identity, it is not so difficult to prove it; we give two proofs, each about half a page long, one of which is based on a variant of the Cauchy-Binet formula , and the other based on properties of the adjugate matrix.

But perhaps it is surprising that such a formula exists at all; one does not normally expect to learn much information about eigenvectors purely from knowledge of eigenvalues. In the random matrix theory literature, for instance in this paper of Erdos, Schlein, and Yau , or this later paper of Van Vu and myself , a related identity has been used, namely. I do so below the fold; we ended up not putting this proof in the note as it was longer than the two other proofs we found. It was certainly something of a surprise to me that there is no explicit appearance of the components of in the formula 1 though they do indirectly appear through their effect on the eigenvalues ; for instance from taking traces one sees that.

One can get some feeling of the identity 1 by considering some special cases. Suppose for instance that is a diagonal matrix with all distinct entries. The upper left entry of is one of the eigenvalues of. If it is equal to , then the eigenvalues of are the other eigenvalues of , and now the left and right-hand sides of 1 are equal to. At the other extreme, if is equal to a different eigenvalue of , then now appears as an eigenvalue of , and both sides of 1 now vanish. More generally, if we order the eigenvalues and , then the Cauchy interlacing inequalities tell us that.

Thus the identity relates the coefficient sizes of an eigenvector with the extent to which the Cauchy interlacing inequalities are sharp. Theorem 1 Multilinear Kakeya estimate Let be a radius. For each , let denote a finite family of infinite tubes in of radius. Assume the following axiom:. Then, for any , one has. It turns out that this machinery is somewhat flexible, and can be used to establish some other estimates of this type. The first result of this paper is to extend the above theorem to the curved setting, in which one localises to a ball of radius and sets to be small , but allows the tubes to be curved in a fashion.

A key point in this approach is that one obtains optimal bounds not losing factors of or , so long as one stays away from the endpoint case which does not seem to be easily treatable by the heat flow methods. Previously, the paper of Bennett, Carbery, and myself was able to use an induction on scale argument to obtain a curved multilinear Kakeya estimate losing a factor of after optimising the argument ; later arguments of Bourgain-Guth and Carbery-Valdimarsson , based on algebraic topology methods, could also obtain a curved multilinear Kakeya estimate without such losses, but only in the algebraic case when the tubes were neighbourhoods of algebraic curves of bounded degree.

Perhaps more interestingly, we are also able to extend the heat flow monotonicity method to apply directly to the multilinear restriction problem, giving the following global multilinear restriction estimate:. Theorem 2 Multilinear restriction theorem Let be an exponent, and let be a parameter.

Let be a sufficiently large natural number, depending only on. For , let be an open subset of , and let be a smooth function obeying the following axioms:. Let be the sets. Local versions of such estimate, in which is replaced with for some , and one accepts a loss of the form , were already established by Bennett, Carbery, and myself using an induction on scale argument. There are two main new ingredients in the proof of Theorem 2. In particular, we actually prove the more general family of estimates.

With logarithmic losses, it is not difficult to establish this estimate by an upward induction on. To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies can be expressed in terms of the magnitude squared of what is essentially the Gabor transform of , and these are non-negative; furthermore, the dispersion relation associated to the extension operators implies that these Gabor transforms propagate along tubes, so that the situation becomes quite similar up to several additional lower order error terms to that in the multilinear Kakeya problem.

This can be viewed as a continuous version of the usual wave packet decomposition method used to relate restriction and Kakeya problems, which when combined with the heat flow monotonicity method allows for one to use a continuous version of induction on scales methods that do not concede any logarithmic factors. Finally, one can combine the curved multilinear Kakeya result with the multilinear restriction result to obtain estimates for multilinear oscillatory integrals away from the endpoint.

Again, this sort of implication was already established in the previous paper of Bennett, Carbery, and myself , but the arguments there had some epsilon losses in the exponents; here we were able to run the argument more carefully and avoid these losses. CA , math. Earlier this month, Hao Huang who, incidentally, was a graduate student here at UCLA gave a remarkably short proof of a long-standing problem in theoretical computer science known as the sensitivity conjecture. See for instance this blog post of Gil Kalai for further discussion and links to many other online discussions of this result.

One formulation of the theorem proved is as follows. Define the -dimensional hypercube graph to be the graph with vertex set , and with every vertex joined to the vertices , where is the standard basis of. Theorem 1 Lower bound on maximum degree of induced subgraphs of hypercube Let be a set of at least vertices in.

Then there is a vertex in that is adjacent in to at least other vertices in. The bound or more precisely, is completely sharp, as shown by Chung, Furedi, Graham, and Seymour ; we describe this example below the fold. When combined with earlier reductions of Gotsman-Linial and Nisan-Szegedy ; we give these below the fold also.

Let be the adjacency matrix of where we index the rows and columns directly by the vertices in , rather than selecting some enumeration , thus when for some , and otherwise. The above theorem then asserts that if is a set of at least vertices, then the minor of has a row or column that contains at least non-zero entries. The key step to prove this theorem is the construction of rather curious variant of the adjacency matrix :.

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Proposition 2 There exists a matrix which is entrywise dominated by in the sense that. Assuming this proposition, the proof of Theorem 1 can now be quickly concluded. If we view as a linear operator on the -dimensional space of functions of , then by hypothesis this space has a -dimensional subspace on which acts by multiplication by. If is a set of at least vertices in , then the space of functions on has codimension at most in , and hence intersects non-trivially.

Thus the minor of also has as an eigenvalue this can also be derived from the Cauchy interlacing inequalities , and in particular this minor has operator norm at least. Remark 3 The argument actually gives a strengthening of Theorem 1 : there exists a vertex of with the property that for every natural number , there are at least paths of length in the restriction of to that start from.

Indeed, if we let be an eigenfunction of on , and let be a vertex in that maximises the value of , then for any we have that the component of is equal to ; on the other hand, by the triangle inequality, this component is at most times the number of length paths in starting from , giving the claim. In previous literature using this method see e. It will be interesting to see what further variants and applications of this method emerge in the near future.

Thanks to Anurag Bishoi in the comments for these references. Very recently, Roman Karasev gave an interpretation of this matrix in terms of the exterior algebra on. In this post I would like to give an alternate interpretation in terms of the operation of twisted convolution , which originated in the theory of the Heisenberg group in quantum mechanics. Firstly note that the original adjacency matrix , when viewed as a linear operator on , is a convolution operator.

As is well known, this operation is commutative and associative. Thus for instance the square of the adjacency operator is also a convolution operator. The factor in this expansion comes from combining the two terms and , which both evaluate to. More generally, given any bilinear form , one can define the twisted convolution. This operation is no longer commutative unless is symmetric. However, it remains associative; indeed, one can easily compute that. For general bilinear forms , this twisted convolution is just as messy as is.

But if we take the specific bilinear form. Thus the only eigenvalues of are and. The matrix is entrywise dominated by in the sense of 1 , and in particular has trace zero; thus the and eigenvalues must occur with equal multiplicity, so in particular the eigenvalue occurs with multiplicity since the matrix has dimensions. This establishes Proposition 2. Remark 4 Twisted convolution is actually just a component of ordinary convolution, but not on the original group ; instead it relates to convolution on a Heisenberg group extension of this group.

More specifically, define the Heisenberg group to be the set of pairs with group law. Convolution on is defined in the usual manner: one has. Now if is a function on the original group , we can define the lift by the formula. Remark 5 With the twisting by the specific bilinear form given by 2 , convolution by and now anticommute rather than commute.

This makes the twisted convolution algebra isomorphic to a Clifford algebra the real or complex algebra generated by formal generators subject to the relations for rather than the commutative algebra more familiar to abelian Fourier analysis. This connection to Clifford algebra also observed independently by Tom Mrowka and by Daniel Matthews may be linked to the exterior algebra interpretation of the argument in the recent preprint of Karasev mentioned above.

Remark 6 One could replace the form 2 in this argument by any other bilinear form that obeyed the relations and for. However, this additional level of generality does not add much; any such will differ from by an antisymmetric form so that for all , which in characteristic two implied that for all , and such forms can always be decomposed as , where. As such, the matrices and are conjugate, with the conjugation operator being the diagonal matrix with entries at each vertex.

Remark 7 Added later This remark combines the two previous remarks. More precisely, let denote the vector space of functions from the hypercube to the Clifford algebra; as a real vector space, this is a dimensional space, isomorphic to the direct sum of copies of , as the Clifford algebra is itself dimensional. One can then define a canonical Clifford adjacency operator on this space by. This operator can either be identified with a Clifford-valued matrix or as a real-valued matrix.

In either case one still has the key algebraic relations and , ensuring that when viewed as a real matrix, half of the eigenvalues are equal to and half equal to. To relate to the real matrices , first observe that each point in the hypercube can be associated with a one-dimensional real subspace i. This can be viewed as a discrete line bundle over the hypercube. Since for any , we see that the -dimensional real linear subspace of of sections of this bundle, that is to say the space of functions such that for all , is an invariant subspace of.

Indeed, using the left-action of the Clifford algebra on , which commutes with , one can naturally identify with , with the left action of acting purely on the first factor and acting purely on the second factor. Any trivialisation of this line bundle lets us interpret the restriction of to as a real matrix.

In particular, given one of the bilinear forms from Remark 6 , we can identify with by identifying any real function with the lift defined by. A somewhat tedious computation using the properties of then eventually gives the intertwining identity. AP , math. CO Tags: fractional dimension , heat flow , multilinear Kakeya conjecture , symmetric polynomials by Terence Tao 20 comments. Let be some domain such as the real numbers. For any natural number , let denote the space of symmetric real-valued functions on variables , thus. For instance, for any natural numbers , the elementary symmetric polynomials.

With the pointwise product operation, becomes a commutative real algebra. We include the case , in which case consists solely of the real constants. Thus for instance. With these conventions, we see that vanishes for , and is equal to if. We also have the transitivity. The lifting map is a linear map from to , but it is not a ring homomorphism.

For instance, when , one has. Combinatorially, the identity 2 follows from the fact that given any injections and with total image of cardinality , one has , and furthermore there exist precisely triples of injections , , such that and. Example 1 When , one has. Note that the coefficients appearing in 2 do not depend on the final number of variables. We may therefore abstract the role of from the law 2 by introducing the real algebra of formal sums.

Thus for instance, in this algebra we have. One can check somewhat tediously that is indeed a commutative real algebra, with a unit. I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule.

For natural numbers , there is an obvious specialisation map from to , defined by the formula. Thus, for instance, maps to and to. From 2 and 3 we see that this map is an algebra homomorphism, even though the maps and are not homomorphisms. By inspecting the component of we see that the homomorphism is in fact surjective. Now suppose that we have a measure on the space , which then induces a product measure on every product space.

To avoid degeneracies we will assume that the integral is strictly positive. Assuming suitable measurability and integrability hypotheses, a function can then be integrated against this product measure to produce a number. Thus for instance, if ,. Note that by hypothesis, only finitely many terms on the right-hand side are non-zero. Now for a key observation: whereas the left-hand side of 6 only makes sense when is a natural number, the right-hand side is meaningful when takes a fractional value or even when it takes negative or complex values!

More precisely, for arbitrary real or complex , we now define to be the space of abstract objects. In particular, the multiplication law 2 continues to hold for such values of , thanks to 3. Thus, for instance, with this formalism the identities 4 , 5 now hold for fractional values of , even though the formal space no longer makes sense as a set, and the formal measure no longer makes sense as a measure. The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals.

See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables and in particular over fermionic variables. Example 2 Suppose is a probability measure on , and is a random variable; on any power , we let be the usual independent copies of on , thus for. Then for any real or complex , the formal integral. For a natural number, this identity has the probabilistic interpretation. One can thus view 7 as an abstract generalisation of 8 to the case when is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about independent copies of in the standard framework of probability theory.

In this particular case, the quantity 7 is non-negative for every nonnegative , which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if has mean zero, one can check that. This is a shame, because otherwise one could hope to start endowing with some sort of commutative von Neumann algebra type structure or the abstract probability structure discussed in this previous post and then interpret it as a genuine measure space rather than as a virtual one.

However, one vestige of positivity remains: if is non-negative, then so is. One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents and appealing to a general extrapolation theorem to then obtain various identities in the fractional case.

The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity. As such, one can manipulate powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is. Lemma 3 Differentiation formula Suppose that a positive measure on depends on some parameter and varies by the formula. Let be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have.

If we allow to now depend on also, then we have the more general total derivative formula. Proof: We just prove 10 , as 11 then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case for some symmetric function for a natural number.

By 6 , the left-hand side of 10 is then. Since , we can write this expression using 6 as. Remark 4 It is also instructive to prove this lemma in the special case when is a natural number, in which case the fractional dimensional integral can be interpreted as a classical integral. In this case, the identity 10 is immediate from applying the product rule to 9 to conclude that. One could in fact derive 10 for arbitrary real or complex from the case when is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details. Proposition 5 Heat flow monotonicity Let be a solution to the heat equation with initial data a rapidly decreasing finite non-negative Radon measure, or more explicitly.

Then for any , the quantity. Proof: By a limiting argument we may assume that is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below. For any , let denote the Radon measure. Then the quantity can be written as a fractional dimensional integral. To simplify this expression we will take advantage of integration by parts in the variable. Specifically, in any direction , we have. Multiplying by and integrating by parts, we see that. Similarly, if is any reasonable function depending only on , we have. The choice of that then achieves the most cancellation turns out to be this cancels the terms that are linear or quadratic in the , so that.

Repeating the calculations establishing 7 , one has. Since , one thus has. This expression is clearly non-negative for , equal to zero for , and positive for , giving the claim. One could simplify here as if desired, though it is not strictly necessary to do so for the proof. Remark 6 As with Remark 4 , one can also establish the identity 14 first for natural numbers by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of. This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of.

However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.

Each author contributes a story of how they encountered some internal or external difficulty in advancing their mathematical career, and how they were able to deal with such difficulties. I myself have contributed one of these essays ; I was initially somewhat surprised when I was approached for a contribution, as my career trajectory has been somewhat of an outlier, and I have been very fortunate to not experience to the same extent many of the obstacles that other contributors write about in this text.

Nevertheless there was a turning point in my career that I write about here during my graduate years, when I found that the improvised and poorly disciplined study habits that were able to get me into graduate school due to an over-reliance on raw mathematical ability were completely inadequate to handle the graduate qualifying exam.

With a combination of an astute advisor and some sheer luck, I was able to pass the exam and finally develop a more sustainable approach to learning and doing mathematics, but it could easily have gone quite differently.

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My 20 year old writeup of this examination, complete with spelling errors, may be found here. CA Tags: asymptotics , decoupling , induction on scales , linear programming , restriction theorems by Terence Tao 11 comments. 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.

In many applications, this bound is nearly tight in the sense that one can easily establish a matching lower bound. 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. 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 inequality. 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. We let be the smallest constant for which one has the restriction inequality. 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. Example 3 Multilinear Kakeya inequality Again ranges over all of.

Let be a fixed dimension, and let be compact subsets of the sphere which are transverse in 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.

Dans la même section

For each , we let be the smallest constant for which one has the multilinear Kakeya inequality. 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 are transverse as in the previous example.

For each , we let be the smallest constant for which one has the multilinear restriction inequality.

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 operators. 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. 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. 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. Then this implies that has a subpolynomial upper bound. Indeed, one can iterate this inequality to show that.

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 7 If is of polynomial size, and obeys the inequality. 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.

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One can then observe the following estimates:. 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. 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.

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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 For any one has. Application of multilinear Kakeya and decoupling If are sufficiently small e. 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. 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. A routine iteration then gives. 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.

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 8 Show that one still obtains a subpolynomial upper bound if the estimate 10 is replaced with. 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.

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To bound the quantity , some auxiliary quantities are introduced for various exponents and and , with the following bounds:. Crude upper bound for For any 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.

The final step iii requires no knowledge of where these quantities come from in harmonic analysis, but the iterations involved can become extremely complicated. 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. Notice how the passage to upper exponents converts the estimate to a simpler inequality. 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. 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 , 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.

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  • 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. If we define. This leads to a contradiction when , and hence as desired. 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. 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.

    In view of the latter two estimates it is natural to restrict attention to the quantities for.