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Suppose is a continuous (but nonlinear) map from one normed vector space to another . The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or , particularly if is not known to be uniformly continuous). If is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

for some depending on , if is small enough; one can of course make independent of (and drop the smallness condition) if is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like in terms of quantities like . There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

This estimate is usually not very good when and are close together. However, when and are far apart, this estimate can be more or less sharp. For instance, if the magnitude of varies so much from to that is more than (say) twice that of , or vice versa, then (1) is sharp up to a multiplicative constant. Also, if is oscillatory in nature, and the distance between and exceeds the “wavelength” of the oscillation of at (or at ), then one also typically expects (1) to be close to sharp. Conversely, if does not vary much in magnitude from to , and the distance between and is less than the wavelength of any oscillation present in , one expects to be able to improve upon (1).

When is relatively simple in form, one can sometimes proceed simply by substituting . For instance, if is the squaring function in a commutative ring , one has

and thus

or in terms of the original variables one has

If the ring is not commutative, one has to modify this to

Thus, for instance, if are matrices and denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity of operator norm that

If involves (or various components of ) in several places, one can sometimes get a good estimate by “swapping” with at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function in a non-commutative ring, we have

which for instance leads to a slight improvement of (2):

More generally, for any natural number , one has the identity

in a commutative ring, while in a non-commutative ring one must modify this to

and for matrices one has

Exercise 1If and are unitary matrices, show that the commutator obeys the inequality(

Hint:first control .)

Now suppose (for simplicity) that is a map between Euclidean spaces. If is continuously differentiable, then one can use the fundamental theorem of calculus to write

where is any continuously differentiable path from to . For instance, if one uses the straight line path , one has

In the one-dimensional case , this simplifies to

Among other things, this immediately implies the factor theorem for functions: if is a function for some that vanishes at some point , then factors as the product of and some function . Another basic consequence is that if is uniformly bounded in magnitude by some constant , then is Lipschitz continuous with the same constant .

Applying (4) to the power function , we obtain the identity

which can be compared with (3). Indeed, for and close to , one can use logarithms and Taylor expansion to arrive at the approximation , so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2For each , let and be random variables taking values in a measurable space , and let be a bounded measurable function.

- (i) (Lindeberg exchange identity) Show that
- (ii) (Knowles-Yin exchange identity) Show that
where is a mixture of and , with uniformly drawn from independently of each other and of the .

- (iii) Discuss the relationship between the identities in parts (i), (ii) with the identities (3), (5).
(The identity in (i) is the starting point for the

Lindeberg exchange methodin probability theory, discussed for instance in this previous post. The identity in (ii) can also be used in the Lindeberg exchange method; the terms in the right-hand side are slightly more symmetric in the indices , which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3If is continuously times differentiable, establish Taylor’s theorem with remainderIf is bounded, conclude that

For real scalar functions , the average value of the continuous real-valued function must be attained at some point in the interval . We thus conclude the mean-value theorem

for some (that can depend on , , and ). This can for instance give a second proof of fact that continuously differentiable functions with bounded derivative are Lipschitz continuous. However it is worth stressing that the mean-value theorem is only available for *real scalar* functions; it is false for instance for complex scalar functions. A basic counterexample is given by the function ; there is no for which . On the other hand, as has magnitude , we still know from (4) that is Lipschitz of constant , and when combined with (1) we obtain the basic bounds

which are already very useful for many applications.

Exercise 4Let be matrices, and let be a non-negative real.

- (i) Establish the Duhamel formula
where denotes the matrix exponential of . (

Hint:Differentiate or in .)- (ii) Establish the
iterated Duhamel formulafor any .

- (iii) Establish the infinitely iterated Duhamel formula
- (iv) If is an matrix depending in a continuously differentiable fashion on , establish the variation formula
where is the adjoint representation applied to , and is the function

(thus for non-zero ), with defined using functional calculus.

We remark that further manipulation of (iv) of the above exercise using the fundamental theorem of calculus eventually leads to the Baker-Campbell-Hausdorff-Dynkin formula, as discussed in this previous blog post.

Exercise 5Let be positive definite matrices, and let be an matrix. Show that there is a unique solution to the Sylvester equationwhich is given by the formula

In the above examples we had applied the fundamental theorem of calculus along linear curves . However, it is sometimes better to use other curves. For instance, the circular arc can be useful, particularly if and are “orthogonal” or “independent” in some sense; a good example of this is the proof by Maurey and Pisier of the gaussian concentration inequality, given in Theorem 8 of this previous blog post. In a similar vein, if one wishes to compare a scalar random variable of mean zero and variance one with a Gaussian random variable of mean zero and variance one, it can be useful to introduce the intermediate random variables (where and are independent); note that these variables have mean zero and variance one, and after coupling them together appropriately they evolve by the Ornstein-Uhlenbeck process, which has many useful properties. For instance, one can use these ideas to establish monotonicity formulae for entropy; see e.g. this paper of Courtade for an example of this and further references. More generally, one can exploit curves that flow according to some geometrically natural ODE or PDE; several examples of this occur famously in Perelman’s proof of the Poincaré conjecture via Ricci flow, discussed for instance in this previous set of lecture notes.

In some cases, it is difficult to compute or the derivative directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map (defined on the open dense subset of matrices consisting of invertible matrices). If one wants to compute for close to , one can write temporarily write , thus

Multiplying both sides on the left by to eliminate the term, and on the right by to eliminate the term, one obtains

and thus on reversing these steps we arrive at the basic identity

For instance, if are matrices, and we consider the resolvents

then we have the *resolvent identity*

as long as does not lie in the spectrum of or (for instance, if , are self-adjoint then one can take to be any strictly complex number). One can iterate this identity to obtain

for any natural number ; in particular, if has operator norm less than one, one has the Neumann series

Similarly, if is a family of invertible matrices that depends in a continuously differentiable fashion on a time variable , then by implicitly differentiating the identity

in using the product rule, we obtain

and hence

(this identity may also be easily derived from (6)). One can then use the fundamental theorem of calculus to obtain variants of (6), for instance by using the curve we arrive at

assuming that the curve stays entirely within the set of invertible matrices. While this identity may seem more complicated than (6), it is more symmetric, which conveys some advantages. For instance, using this identity it is easy to see that if are positive definite with in the sense of positive definite matrices (that is, is positive definite), then . (Try to prove this using (6) instead!)

Exercise 6If is an invertible matrix and are vectors, establish the Sherman-Morrison formulawhenever is a scalar such that is non-zero. (See also this previous blog post for more discussion of these sorts of identities.)

One can use the Cauchy integral formula to extend these identities to other functions of matrices. For instance, if is an entire function, and is a counterclockwise contour that goes around the spectrum of both and , then we have

and similarly

and hence by (7) one has

similarly, if depends on in a continuously differentiable fashion, then

as long as goes around the spectrum of .

Exercise 7If is an matrix depending continuously differentiably on , and is an entire function, establish the tracial chain rule

In a similar vein, given that the logarithm function is the antiderivative of the reciprocal, one can express the matrix logarithm of a positive definite matrix by the fundamental theorem of calculus identity

(with the constant term needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if is a family of positive definite matrices depending continuously on , that

This can be used for instance to show that is a monotone increasing function, in the sense that whenever in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference of the logarithm of two positive definite matrices .

To compare the square root of two positive definite matrices is trickier; there are multiple ways to proceed. One approach is to use contour integration as before (but one has to take some care to avoid branch cuts of the square root). Another to express the square root in terms of exponentials via the formula

where is the gamma function; this formula can be verified by first diagonalising to reduce to the scalar case and using the definition of the Gamma function. Then one has

and one can use some of the previous identities to control . This is pretty messy though. A third way to proceed is via implicit differentiation. If for instance is a family of positive definite matrices depending continuously differentiably on , we can differentiate the identity

to obtain

This can for instance be solved using Exercise 5 to obtain

and this can in turn be integrated to obtain a formula for . This is again a rather messy formula, but it does at least demonstrate that the square root is a monotone increasing function on positive definite matrices: implies .

Several of the above identities for matrices can be (carefully) extended to operators on Hilbert spaces provided that they are sufficiently well behaved (in particular, if they have a good functional calculus, and if various spectral hypotheses are obeyed). We will not attempt to do so here, however.

Suppose one has a bounded sequence of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit , which in this post I will refer to as the *classical limit* to distinguish from the other limits discussed in this post. The classical limit, if it exists, is the unique real number such that for every , one has for all sufficiently large . We say that a sequence is (classically) convergent if its classical limit exists. The classical limit obeys many useful *limit laws* when applied to classically convergent sequences. Firstly, it is linear: if and are classically convergent sequences, then is also classically convergent with

and similarly for any scalar , is classically convergent with

It is also an algebra homomorphism: is also classically convergent with

We also have shift invariance: if is classically convergent, then so is with

and more generally in fact for any injection , is classically convergent with

The classical limit of a sequence is unchanged if one modifies any finite number of elements of the sequence. Finally, we have boundedness: for any classically convergent sequence , one has

One can in fact show without much difficulty that these laws uniquely determine the classical limit functional on convergent sequences.

One would like to extend the classical limit notion to more general bounded sequences; however, when doing so one must give up one or more of the desirable limit laws that were listed above. Consider for instance the sequence . On the one hand, one has for all , so if one wishes to retain the homomorphism property (3), any “limit” of this sequence would have to necessarily square to , that is to say it must equal or . On the other hand, if one wished to retain the shift invariance property (4) as well as the homogeneity property (2), any “limit” of this sequence would have to equal its own negation and thus be zero.

Nevertheless there are a number of useful generalisations and variants of the classical limit concept for non-convergent sequences that obey a significant portion of the above limit laws. For instance, we have the limit superior

and limit inferior

which are well-defined real numbers for any bounded sequence ; they agree with the classical limit when the sequence is convergent, but disagree otherwise. They enjoy the shift-invariance property (4), and the boundedness property (6), but do not in general obey the homomorphism property (3) or the linearity property (1); indeed, we only have the subadditivity property

for the limit superior, and the superadditivity property

for the limit inferior. The homogeneity property (2) is only obeyed by the limits superior and inferior for non-negative ; for negative , one must have the limit inferior on one side of (2) and the limit superior on the other, thus for instance

The limit superior and limit inferior are examples of limit points of the sequence, which can for instance be defined as points that are limits of at least one subsequence of the original sequence. Indeed, the limit superior is always the largest limit point of the sequence, and the limit inferior is always the smallest limit point. However, limit points can be highly non-unique (indeed they are unique if and only if the sequence is classically convergent), and so it is difficult to sensibly interpret most of the usual limit laws in this setting, with the exception of the homogeneity property (2) and the boundedness property (6) that are easy to state for limit points.

Another notion of limit are the Césaro limits

if this limit exists, we say that the sequence is Césaro convergent. If the sequence already has a classical limit, then it also has a Césaro limit that agrees with the classical limit; but there are additional sequences that have a Césaro limit but not a classical one. For instance, the non-classically convergent sequence discussed above is Césaro convergent, with a Césaro limit of . However, there are still bounded sequences that do not have Césaro limit, such as (exercise!), basically because such sequences oscillate too slowly for the Césaro averaging to be of much use in accelerating the convergence. The Césaro limit is linear, bounded, and shift invariant, but not an algebra homomorphism and also does not obey the rearrangement property (5).

Using the Hahn-Banach theorem, one can extend the classical limit functional to *generalised limit functionals* , defined to be bounded linear functionals from the space of bounded real sequences to the real numbers that extend the classical limit functional (defined on the space of convergent sequences) without any increase in the operator norm. (In some of my past writings I made the slight error of referring to these generalised limit functionals as Banach limits, though as discussed below, the latter actually refers to a subclass of generalised limit functionals.) It is not difficult to see that such generalised limit functionals will range between the limit inferior and limit superior. In fact, for any specific sequence and any number lying in the closed interval , there exists at least one generalised limit functional that takes the value when applied to ; for instance, for any number in , there exists a generalised limit functional that assigns that number as the “limit” of the sequence . This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by , and then appealing to the Hahn-Banach theorem to extend to all sequences. This observation also gives a necessary and sufficient criterion for a bounded sequence to classically converge to a limit , namely that all generalised limits of this sequence must equal .

~~Because of the reliance on the Hahn-Banach theorem, the existence of generalised limits requires the axiom of choice (or some weakened version thereof); there are models of set theory without the axiom of choice in which no generalised limits exist. For instance, consider a Solovay model in which all subsets of the real numbers are measurable. If one lets denote the function that extracts the ternary digit past the decimal point (thus , and lets be a generalised limit functional, then the function is non-constant (e.g. and ), but also invariant almost everywhere with respect to translation by ternary rationals , and hence cannot be measurable (due to the continuity of translation in the strong operator topology, or the Steinhaus lemma), and so generalised limit functionals cannot exist.~~

Generalised limits can obey the shift-invariance property (4) or the algebra homomorphism property (3), but as the above analysis of the sequence shows, they cannot do both. Generalised limits that obey the shift-invariance property (4) are known as Banach limits; one can for instance construct them by applying the Hahn-Banach theorem to the Césaro limit functional; alternatively, if is any generalised limit, then the Césaro-type functional will be a Banach limit. The existence of Banach limits can be viewed as a demonstration of the amenability of the natural numbers (or integers); see this previous blog post for further discussion.

Generalised limits that obey the algebra homomorphism property (3) are known as *ultrafilter limits*. If one is given a generalised limit functional that obeys (3), then for any subset of the natural numbers , the generalised limit must equal its own square (since ) and is thus either or . If one defines to be the collection of all subsets of for which , one can verify that obeys the axioms of a non-principal ultrafilter. Conversely, if is a non-principal ultrafilter, one can define the associated generalised limit of any bounded sequence to be the unique real number such that the sets lie in for all ; one can check that this does indeed give a well-defined generalised limit that obeys (3). Non-principal ultrafilters can be constructed using Zorn’s lemma. In fact, they do not quite need the full strength of the axiom of choice; see the Wikipedia article on the ultrafilter lemma for examples.

We have previously noted that generalised limits of a sequence can converge to any point between the limit inferior and limit superior. The same is not true if one restricts to Banach limits or ultrafilter limits. For instance, by the arguments already given, the only possible Banach limit for the sequence is zero. Meanwhile, an ultrafilter limit must converge to a limit point of the original sequence, but conversely every limit point can be attained by at least one ultrafilter limit; we leave these assertions as an exercise to the interested reader. In particular, a bounded sequence converges classically to a limit if and only if all ultrafilter limits converge to .

There is no generalisation of the classical limit functional to any space that includes non-classically convergent sequences that obeys the subsequence property (5), since any non-classically-convergent sequence will have one subsequence that converges to the limit superior, and another subsequence that converges to the limit inferior, and one of these will have to violate (5) since the limit superior and limit inferior are distinct. So the above limit notions come close to the best generalisations of limit that one can use in practice.

(Added after comments) If denotes the Stone-Cech compactification of the natural numbers, then can be canonically identified with the continuous functions on , and hence by the Riesz representation theorem, bounded linear functionals on can be identified with finite measures on this space. From this it is not difficult to show that generalised limit functionals can be canonically identified with probability measures on the compact Hausdorff space , that ultrafilter limits correspond to those probability measures that are Dirac measures (i.e. they can be canonically identified with points in ), and Banach limits correspond to those probability measures that are invariant with respect to the translation action of the integers on .

We summarise (some of) the above discussion in the following table:

Limit | Always defined | Linear | Shift-invariant | Homomorphism | Constructive |

Classical | No | Yes | Yes | Yes | Yes |

Superior | Yes | No | Yes | No | Yes |

Inferior | Yes | No | Yes | No | Yes |

Césaro | No | Yes | Yes | No | Yes |

Generalised | Yes | Yes | Depends | Depends | No |

Banach | Yes | Yes | Yes | No | No |

Ultrafilter | Yes | Yes | No | Yes | No |

Ben Green and I have (finally!) uploaded to the arXiv our paper “New bounds for Szemerédi’s theorem, III: A polylogarithmic bound for “, submitted to Mathematika. This is the sequel to two previous papers (and an erratum to the former paper), concerning quantitative versions of Szemerédi’s theorem in the case of length four progressions. This sequel has been delayed for over a decade for a number of reasons, but we have finally managed to write the arguments up to our satisfaction and submit it (to a special issue of Mathematika honouring the work of Klaus Roth).

For any natural number , define to be the largest cardinality of a subset of which does not contain any non-trivial arithmetic progressions of length four (where “non-trivial” means that is non-zero). Trivially we have . In 1969, Szemerédi showed that . However, the decay rate that could be theoretically extracted from this argument (and from several subsequent proofs of this bound, including one by Roth) were quite poor. The first significant quantitative bound on this quantity was by Gowers, who showed that for some absolute constant . In the second paper in the above-mentioned series, we managed to improve this bound to . In this paper, we improve the bound further to , which seems to be the limit of the methods. (We remark that if we could take to be larger than one, this would imply the length four case of a well known conjecture of Erdös that any set of natural numbers whose sum of reciprocals diverges would contain arbitrarily long arithmetic progressions. Thanks to the work of Sanders and of Bloom, the corresponding case of the conjecture for length three conjectures is nearly settled, as it is known that for the analogous bound on one can take any less than one.)

Most of the previous work on bounding relied in some form or another on the *density increment argument* introduced by Roth back in 1953; roughly speaking, the idea is to show that if a dense subset of fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of in which has increased density. This was the basic method for instance underlying our previous bound , as well as a finite field analogue of the bound ; however we encountered significant technical difficulties for several years in extending this argument to obtain the result of the current paper. Our method is instead based on “energy increment arguments”, and more specifically on establishing quantitative version of a Khintchine-type recurrence theorem, similar to the qualitative recurrence theorems established (in the ergodic theory context) by Bergelson-Host-Kra, and (in the current combinatorial context) by Ben Green and myself.

One way to phrase the latter recurrence theorem is as follows. Suppose that has density . Then one would expect a “randomly” selected arithmetic progression in (using the convention that random variables will be in boldface) to be contained in with probability about . This is not true in general, however it was shown by Ben and myself that for any , there was a set of shifts of cardinality , such that for any such one had

if was chosen uniformly at random from . This easily implies that , but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take to be extremely large compared to to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift .

We do not know how to improve the lower bound on the set of shifts to the point where it can give bounds that are competitive with those in this paper. However, we can obtain better quantitative results if we permit ourselves to *couple* together the two parameters and of the length four progression. Namely, with , , as above, we are able to show that there exist random variables , not necessarily independent, such that

and such that we have the non-degeneracy bound

This then easily implies the main theorem.

The energy increment method is then deployed to locate a good pair of random variables that will obey the above bounds. One can get some intuition on how to proceed here by considering some model cases. Firstly one can consider a “globally quadratically structured” case in which the indicator function “behaves like” a globally quadratic function such as , for some irrational and some smooth periodic function of mean . If one then takes to be uniformly distributed in and respectively for some small , with no coupling between the two variables, then the left-hand side of (1) is approximately of the form

where the integral is with respect to the probability Haar measure, and the constraint ultimately arises from the algebraic constraint

However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least , which (morally at least) gives (1) in this case.

Due to the nature of the energy increment argument, it also becomes necessary to consider “locally quadratically structured” cases, in which is partitioned into some number of structured pieces (think of these as arithmetic progressions, or as “Bohr sets), and on each piece , behaves like a locally quadratic function such as , where now varies with , and the mean of will be approximately on the average after averaging in (weighted by the size of the pieces ). Now one should select and in the following coupled manner: first one chooses uniformly from , then one defines to be the label such that , and then selects uniformly from a set which is related to in much the same way that is related to . If one does this correctly, the analogue of (2) becomes

and one can again use Cauchy-Schwarz and Fubini’s theorem to conclude.

The general case proceeds, very roughly, by an iterative argument. At each stage of the iteration, one has some sort of quadratic model of which involves a decomposition of into structured pieces , and a quadratic approximation to on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm of the error is small enough) to model the count in (1) (for random variables determined by the above partition of into pieces ), and if the frequencies (such as ) involved in the quadratic approximation are “high rank” or “linearly independent over the rationals” in a suitably quantitative sense, then some version of the above arguments can be made to work. If there are some unwanted linear dependencies in the frequencies, we can do some linear algebra to eliminate one of the frequencies (using some geometry of numbers to keep the quantitative bounds under control) and continue the iteration. If instead the approximation is too inaccurate, then the error will be large in a certain averaged local Gowers uniformity norm . A significant fraction of the paper is then devoted to establishing a quantitative *inverse theorem* for that norm that concludes (with good bounds) that the error must then locally correlate with locally quadratic phases, which can be used to refine the quadratic approximation to in a manner that significantly increases its “energy” (basically an norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.

There are existing inverse theorems for type norms in the literature, going back to the pioneering work of Gowers mentioned previously, and relying on arithmetic combinatorics tools such as Freiman’s theorem and the Balog-Szemerédi-Gowers lemma, which are good for analysing the “-structured homomorphisms” that arise in Gowers’ argument. However, when we applied these methods to the local Gowers norms we obtained inferior quantitative results that were not strong enough for our application. Instead, we use arguments from a different paper of Gowers in which he tackled Szemerédi’s theorem for arbitrary length progressions. This method produces “-structured homomorphisms” associated to any function with large Gowers uniformity norm; however the catch is that such homomorphisms are initially supported only on a sparse unstructured set, rather than a structured set such as a Bohr set. To proceed further, one first has to locate inside the sparse unstructured set a sparse *pseudorandom* subset of a Bohr set, and then use “error-correction” type methods (such as “majority-vote” based algorithms) to locally upgrade this -structured homomorphism on pseudorandom subsets of Bohr sets to a -structured homomorphism on the entirety of a Bohr set. It is then possible to use some “approximate cohomology” tools to “integrate” these homomorphisms (and discern a key “local symmetry” property of these homomorphisms) to locate the desired local quadratic structure (in much the same fashion that a -form on that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the -form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.

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