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 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.
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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.
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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. 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.source
The Decomposition of Figures Into Smaller Parts
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.
By the axioms 22 , these quantities are of the form. We can then eliminate the role of by taking another limit superior. 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. From the convexity 25 and a brief calculation we have. Thus from 29 we have. Finally, 26 gives. We can then eliminate the variables one by one. Inserting 33 into 32 we obtain.
In particular. 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 reads. 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. 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.
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 — Consider the weighted sum.
I 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. Blog at WordPress.
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What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Then we have Conjecture 1 Collatz conjecture One has for all. For instance, it is a result of Krasikov and Lagarias that for all sufficiently large. In this paper we obtain the following further improvement at the cost of weakening natural density to logarithmic density : Theorem 2 Let be any function with.
For instance, one can compute that for large random odd , will take the residue classes with probabilities respectively. A first hint of how to proceed comes from the elementary number theory observation easily proven by induction that the rational numbers are all distinct as vary over tuples in. However, if one groups adjacent terms in 1 together, one can rewrite it assuming is even for sake of discussion as where.
One can use this to establish the bound 4 as long as one can show that the expression is not close to an integer for a moderately large number , to be precise of indices. To put it another way, if we let denote the set of pairs for which we have to show that with overwhelming probability the random walk which we view as a two-dimensional renewal process contains at least a few points lying outside of.
Among other things, Gallagher showed that if the Hardy-Littlewood conjecture was true, then the prime gaps with were asymptotically distributed according to an exponential distribution of mean , in the sense that as for any fixed. The precise results are technical, but roughly we show results of the following form: Theorem 3 Large gaps from Hardy-Littlewood, rough statement If the Hardy-Littlewood conjecture is uniformly true for -tuples of length , and with shifts of size , with a power savings in the error term, then.
One can interpret probabilistically as where is the randomly sifted set of integers formed by removing one residue class uniformly at random for each prime. As stated above, we are able to show that almost surely one has and that the Hardy-Littlewood conjectures hold with power savings for up to for any fixed and for shifts of size.
In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. Here are some examples not an exhaustive list of such blowup criteria: Leray blowup criterion, If blows up at a finite time , and , then for an absolute constant. Prodi — Serrin — Ladyzhenskaya blowup criterion, If blows up at a finite time , and , then , where.
Beale-Kato-Majda blowup criterion, If blows up at a finite time , then , where is the vorticity. Kato blowup criterion, If blows up at a finite time , then for some absolute constant. Escauriaza-Seregin-Sverak blowup criterion, If blows up at a finite time , then. Seregin blowup criterion, If blows up at a finite time , then. Phuc blowup criterion, If blows up at a finite time , then for any.
Gallagher-Koch-Planchon blowup criterion, If blows up at a finite time , then for any. Albritton blowup criterion, If blows up at a finite time , then for any. As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to for some absolute constant , which to my knowledge is the first very slightly supercritical blowup criterion for Navier-Stokes in the literature. The main task is to show that the dimensionless quantity cannot get too large; in particular, we end up establishing a bound of the form from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.
As is well known, the vorticity obeys the vorticity equation In the epoch of regularity , the coefficients of this equation obey good bounds, allowing the machinery of Carleman estimates to come into play. 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.
Eigenvectors from eigenvalues 13 August, in math. The identity is as follows: Theorem 1 Let be an Hermitian matrix, with eigenvalues. Then where is the Hermitian matrix formed by deleting the row and column from. For instance, if we have for some real number , -dimensional vector , and Hermitian matrix , then we have assuming that the denominator is non-zero. 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 but it is not immediately obvious that one can derive the former identity from the latter.
More generally, if we order the eigenvalues and , then the Cauchy interlacing inequalities tell us that for , and for , so that the right-hand side of 1 lies between and , which is of course consistent with 1 as is a unit vector. Assume the following axiom: i Transversality whenever is oriented in the direction of a unit vector for , we have for some , where we use the usual Euclidean norm on the wedge product.
Then, for any , one has where are the usual Lebesgue norms with respect to Lebesgue measure, denotes the indicator function of , and denotes the cardinality of. 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. For , let be an open subset of , and let be a smooth function obeying the following axioms: i Regularity For each and , one has for all.
Related The Decomposition of Figures Into Smaller Parts (Popular Lectures in Mathematics)
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