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Suppose ${F: X \rightarrow Y}$ is a continuous (but nonlinear) map from one normed vector space ${X}$ to another ${Y}$. The continuity means, roughly speaking, that if ${x_0, x \in X}$ are such that ${\|x-x_0\|_X}$ is small, then ${\|F(x)-F(x_0)\|_Y}$ is also small (though the precise notion of “smallness” may depend on ${x}$ or ${x_0}$, particularly if ${F}$ is not known to be uniformly continuous). If ${F}$ is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

$\displaystyle \|F(x)-F(x_0)\|_Y \leq C(x_0) \|x-x_0\|_X$

for some ${C(x_0)}$ depending on ${x_0}$, if ${\|x-x_0\|_X}$ is small enough; one can of course make ${C(x_0)}$ independent of ${x_0}$ (and drop the smallness condition) if ${F}$ is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like ${\|F(x)-F(x_0)\|_Y}$ in terms of quantities like ${\|x-x_0\|_X}$. There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

$\displaystyle \|F(x)-F(x_0)\|_Y \leq \|F(x)\|_Y + \|F(x_0)\|_Y. \ \ \ \ \ (1)$

This estimate is usually not very good when ${x}$ and ${x_0}$ are close together. However, when ${x}$ and ${x_0}$ are far apart, this estimate can be more or less sharp. For instance, if the magnitude of ${F}$ varies so much from ${x_0}$ to ${x}$ that ${\|F(x)\|_Y}$ is more than (say) twice that of ${\|F(x_0)\|_Y}$, or vice versa, then (1) is sharp up to a multiplicative constant. Also, if ${F}$ is oscillatory in nature, and the distance between ${x}$ and ${x_0}$ exceeds the “wavelength” of the oscillation of ${F}$ at ${x_0}$ (or at ${x}$), then one also typically expects (1) to be close to sharp. Conversely, if ${F}$ does not vary much in magnitude from ${x_0}$ to ${x}$, and the distance between ${x}$ and ${x_0}$ is less than the wavelength of any oscillation present in ${F}$, one expects to be able to improve upon (1).

When ${F}$ is relatively simple in form, one can sometimes proceed simply by substituting ${x = x_0 + h}$. For instance, if ${F: R \rightarrow R}$ is the squaring function ${F(x) = x^2}$ in a commutative ring ${R}$, one has

$\displaystyle F(x_0+h) = (x_0+h)^2 = x_0^2 + 2x_0 h+ h^2$

and thus

$\displaystyle F(x_0+h) - F(x_0) = 2x_0 h + h^2$

or in terms of the original variables ${x, x_0}$ one has

$\displaystyle F(x) - F(x_0) = 2 x_0 (x-x_0) + (x-x_0)^2.$

If the ring ${R}$ is not commutative, one has to modify this to

$\displaystyle F(x) - F(x_0) = x_0 (x-x_0) + (x-x_0) x_0 + (x-x_0)^2.$

Thus, for instance, if ${A, B}$ are ${n \times n}$ matrices and ${\| \|_{op}}$ denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity ${\| AB\|_{op} \leq \| A \|_{op} \|B\|_{op}}$ of operator norm that

$\displaystyle \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( 2 \|B\|_{op} + \|A - B \|_{op} ). \ \ \ \ \ (2)$

If ${F(x)}$ involves ${x}$ (or various components of ${x}$) in several places, one can sometimes get a good estimate by “swapping” ${x}$ with ${x_0}$ at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function ${F(x) = x^2 = x x}$ in a non-commutative ring, we have

$\displaystyle F(x) - F(x_0) = x x - x_0 x_0$

$\displaystyle = (x x - x_0 x) + (x_0 x - x_0 x_0)$

$\displaystyle = (x-x_0) x + x_0 (x-x_0)$

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

$\displaystyle \| A^2 - B^2 \|_{op} \leq \| A - B \|_{op} ( \| A\|_{op} + \|B\|_{op} ).$

More generally, for any natural number ${n}$, one has the identity

$\displaystyle x^n - x_0^n = (x-x_0) (x^{n-1} + x^{n-2} x_0 + \dots + x x_0^{n-2} + x_0^{n-1}) \ \ \ \ \ (3)$

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

$\displaystyle x^n - x_0^n = \sum_{i=0}^{n-1} x_0^i (x-x_0) x^{n-1-i},$

and for matrices one has

$\displaystyle \| A^n - B^n \|_{op} \leq \| A-B\|_{op} ( \|A\|_{op}^{n-1} + \| A\|_{op}^{n-2} \| B\|_{op} + \dots + \|B\|_{op}^{n-1} ).$

Exercise 1 If ${U}$ and ${V}$ are unitary ${n \times n}$ matrices, show that the commutator ${[U,V] := U V U^{-1} V^{-1}}$ obeys the inequality

$\displaystyle \| [U,V] - I \|_{op} \leq 2 \| U - I \|_{op} \| V - I \|_{op}.$

(Hint: first control ${\| UV - VU \|_{op}}$.)

Now suppose (for simplicity) that ${F: {\bf R}^d \rightarrow {\bf R}^{d'}}$ is a map between Euclidean spaces. If ${F}$ is continuously differentiable, then one can use the fundamental theorem of calculus to write

$\displaystyle F(x) - F(x_0) = \int_0^1 \frac{d}{dt} F( \gamma(t) )\ dt$

where ${\gamma: [0,1] \rightarrow Y}$ is any continuously differentiable path from ${x_0}$ to ${x}$. For instance, if one uses the straight line path ${\gamma(t) := (1-t) x_0 + tx}$, one has

$\displaystyle F(x) - F(x_0) = \int_0^1 ((x-x_0) \cdot \nabla F)( (1-t) x_0 + t x )\ dt.$

In the one-dimensional case ${d=1}$, this simplifies to

$\displaystyle F(x) - F(x_0) = (x-x_0) \int_0^1 F'( (1-t) x_0 + t x )\ dt. \ \ \ \ \ (4)$

Among other things, this immediately implies the factor theorem for ${C^k}$ functions: if ${F}$ is a ${C^k({\bf R})}$ function for some ${k \geq 1}$ that vanishes at some point ${x_0}$, then ${F(x)}$ factors as the product of ${x-x_0}$ and some ${C^{k-1}}$ function ${G}$. Another basic consequence is that if ${\nabla F}$ is uniformly bounded in magnitude by some constant ${C}$, then ${F}$ is Lipschitz continuous with the same constant ${C}$.

Applying (4) to the power function ${x \mapsto x^n}$, we obtain the identity

$\displaystyle x^n - x_0^n = n (x-x_0) \int_0^1 ((1-t) x_0 + t x)^{n-1}\ dt \ \ \ \ \ (5)$

which can be compared with (3). Indeed, for ${x_0}$ and ${x}$ close to ${1}$, one can use logarithms and Taylor expansion to arrive at the approximation ${((1-t) x_0 + t x)^{n-1} \approx x_0^{(1-t) (n-1)} x^{t(n-1)}}$, so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2 For each ${i=1,\dots,n}$, let ${X^{(1)}_i}$ and ${X^{(0)}_i}$ be random variables taking values in a measurable space ${R_i}$, and let ${F: R_1 \times \dots \times R_n \rightarrow {\bf R}^m}$ be a bounded measurable function.

• (i) (Lindeberg exchange identity) Show that

$\displaystyle \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)$

$\displaystyle = \sum_{i=1}^n \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(1)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n)$

$\displaystyle - \mathop{\bf E} F( X^{(1)}_1,\dots, X^{(1)}_{i-1}, X^{(0)}_i, X^{(0)}_{i+1}, \dots, X^{(0)}_n).$

• (ii) (Knowles-Yin exchange identity) Show that

$\displaystyle \mathop{\bf E} F(X^{(1)}_1,\dots,X^{(1)}_n) - \mathop{\bf E} F(X^{(0)}_1,\dots,X^{(0)}_n)$

$\displaystyle = \int_0^1 \sum_{i=1}^n \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(1)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)$

$\displaystyle - \mathop{\bf E} F( X^{(t)}_1,\dots, X^{(t)}_{i-1}, X^{(0)}_i, X^{(t)}_{i+1}, \dots, X^{(t)}_n)\ dt,$

where ${X^{(t)}_i = 1_{I_i \leq t} X^{(0)}_i + 1_{I_i > t} X^{(1)}_i}$ is a mixture of ${X^{(0)}_i}$ and ${X^{(1)}_i}$, with ${I_1,\dots,I_n}$ uniformly drawn from ${[0,1]}$ independently of each other and of the ${X^{(0)}_1,\dots,X^{(0)}_n, X^{(1)}_0,\dots,X^{(1)}_n}$.

• (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 method in 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 ${1,\dots,n}$, which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3 If ${F: {\bf R}^d \rightarrow {\bf R}^{d'}}$ is continuously ${k}$ times differentiable, establish Taylor’s theorem with remainder

$\displaystyle F(x) = \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )$

$\displaystyle + \int_0^1 \frac{(1-t)^{k-1}}{(k-1)!} (((x-x_0) \cdot \nabla)^k F)((1-t) x_0 + t x)\ dt.$

If ${\nabla^k F}$ is bounded, conclude that

$\displaystyle |F(x) - \sum_{j=0}^{k-1} \frac{1}{j!} (((x-x_0) \cdot \nabla)^j F)( x_0 )|$

$\displaystyle \leq \frac{|x-x_0|^k}{k!} \sup_{y \in {\bf R}^d} |\nabla^k F(y)|.$

For real scalar functions ${F: {\bf R}^d \rightarrow {\bf R}}$, the average value of the continuous real-valued function ${(x - x_0) \cdot \nabla F((1-t) x_0 + t x)}$ must be attained at some point ${t}$ in the interval ${[0,1]}$. We thus conclude the mean-value theorem

$\displaystyle F(x) - F(x_0) = ((x - x_0) \cdot \nabla F)((1-t) x_0 + t x)$

for some ${t \in [0,1]}$ (that can depend on ${x}$, ${x_0}$, and ${F}$). This can for instance give a second proof of fact that continuously differentiable functions ${F}$ 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 ${e(x) := e^{2\pi i x}}$; there is no ${t \in [0,1]}$ for which ${e(1) - e(0) = e'(t)}$. On the other hand, as ${e'}$ has magnitude ${2\pi}$, we still know from (4) that ${e}$ is Lipschitz of constant ${2\pi}$, and when combined with (1) we obtain the basic bounds

$\displaystyle |e(x) - e(y)| \leq \min( 2, 2\pi |x-y| )$

which are already very useful for many applications.

Exercise 4 Let ${H_0, V}$ be ${n \times n}$ matrices, and let ${t}$ be a non-negative real.

• (i) Establish the Duhamel formula

$\displaystyle e^{t(H_0+V)} = e^{tH_0} + \int_0^t e^{(t-s) H_0} V e^{s (H_0+V)}\ ds$

$\displaystyle = e^{tH_0} + \int_0^t e^{(t-s) (H_0+V)} V e^{s H_0}\ ds$

where ${e^A}$ denotes the matrix exponential of ${A}$. (Hint: Differentiate ${e^{(t-s) H_0} e^{s (H_0+V)}}$ or ${e^{(t-s) (H_0+V)} e^{s H_0}}$ in ${s}$.)

• (ii) Establish the iterated Duhamel formula

$\displaystyle e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^k \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}$

$\displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j$

$\displaystyle + \int_{0 \leq t_1 \leq \dots \leq t_{k+1} \leq t}$

$\displaystyle e^{(t-t_{k+1}) (H_0+V)} V e^{(t_{k+1}-t_k) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_{k+1}$

for any ${k \geq 0}$.

• (iii) Establish the infinitely iterated Duhamel formula

$\displaystyle e^{t(H_0+V)} = e^{tH_0} + \sum_{j=1}^\infty \int_{0 \leq t_1 \leq \dots \leq t_j \leq t}$

$\displaystyle e^{(t-t_j) H_0} V e^{(t_j-t_{j-1}) H_0} V \dots e^{(t_2-t_1) H_0} V e^{t_1 H_0}\ dt_1 \dots dt_j.$

• (iv) If ${H(t)}$ is an ${n \times n}$ matrix depending in a continuously differentiable fashion on ${t}$, establish the variation formula

$\displaystyle \frac{d}{dt} e^{H(t)} = (F(\mathrm{ad}(H(t))) H'(t)) e^{H(t)}$

where ${\mathrm{ad}(H)}$ is the adjoint representation ${\mathrm{ad}(H)(V) = HV - VH}$ applied to ${H}$, and ${F}$ is the function

$\displaystyle F(z) := \int_0^1 e^{sz}\ ds$

(thus ${F(z) = \frac{e^z-1}{z}}$ for non-zero ${z}$), with ${F(\mathrm{ad}(H(t)))}$ 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 5 Let ${A, B}$ be positive definite ${n \times n}$ matrices, and let ${Y}$ be an ${n \times n}$ matrix. Show that there is a unique solution ${X}$ to the Sylvester equation

$\displaystyle AX + X B = Y$

which is given by the formula

$\displaystyle X = \int_0^\infty e^{-tA} Y e^{-tB}\ dt.$

In the above examples we had applied the fundamental theorem of calculus along linear curves ${\gamma(t) = (1-t) x_0 + t x}$. However, it is sometimes better to use other curves. For instance, the circular arc ${\gamma(t) = \cos(\pi t/2) x_0 + \sin(\pi t/2) x}$ can be useful, particularly if ${x_0}$ and ${x}$ 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 ${X}$ of mean zero and variance one with a Gaussian random variable ${G}$ of mean zero and variance one, it can be useful to introduce the intermediate random variables ${\gamma(t) := (1-t)^{1/2} X + t^{1/2} G}$ (where ${X}$ and ${G}$ 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 ${\gamma}$ 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 ${F(x)-F(x_0)}$ or the derivative ${\nabla F}$ directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map ${F(A) := A^{-1}}$ (defined on the open dense subset of ${n \times n}$ matrices consisting of invertible matrices). If one wants to compute ${F(B)-F(A)}$ for ${B}$ close to ${A}$, one can write temporarily write ${F(B) - F(A) = E}$, thus

$\displaystyle B^{-1} - A^{-1} = E.$

Multiplying both sides on the left by ${B}$ to eliminate the ${B^{-1}}$ term, and on the right by ${A}$ to eliminate the ${A^{-1}}$ term, one obtains

$\displaystyle A - B = B E A$

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

$\displaystyle B^{-1} - A^{-1} = B^{-1} (A - B) A^{-1}. \ \ \ \ \ (6)$

For instance, if ${H_0, V}$ are ${n \times n}$ matrices, and we consider the resolvents

$\displaystyle R_0(z) := (H_0 - z I)^{-1}; \quad R_V(z) := (H_0 + V - zI)^{-1}$

then we have the resolvent identity

$\displaystyle R_V(z) - R_0(z) = - R_V(z) V R_0(z) \ \ \ \ \ (7)$

as long as ${z}$ does not lie in the spectrum of ${H_0}$ or ${H_0+V}$ (for instance, if ${H_0}$, ${V}$ are self-adjoint then one can take ${z}$ to be any strictly complex number). One can iterate this identity to obtain

$\displaystyle R_V(z) = \sum_{j=0}^k (-R_0(z) V)^j R_0(z) + (-R_V(z) V) (-R_0(z) V)^k R_0(z)$

for any natural number ${k}$; in particular, if ${R_0(z) V}$ has operator norm less than one, one has the Neumann series

$\displaystyle R_V(z) = \sum_{j=0}^\infty (-R_0(z) V)^j R_0(z).$

Similarly, if ${A(t)}$ is a family of invertible matrices that depends in a continuously differentiable fashion on a time variable ${t}$, then by implicitly differentiating the identity

$\displaystyle A(t) A(t)^{-1} = I$

in ${t}$ using the product rule, we obtain

$\displaystyle (\frac{d}{dt} A(t)) A(t)^{-1} + A(t) \frac{d}{dt} A(t)^{-1} = 0$

and hence

$\displaystyle \frac{d}{dt} A(t)^{-1} = - A(t)^{-1} (\frac{d}{dt} A(t)) A(t)^{-1}$

(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 ${\gamma(t) = (1-t) A + tB}$ we arrive at

$\displaystyle B^{-1} - A^{-1} = \int_0^1 ((1-t)A + tB)^{-1} (A-B) ((1-t)A + tB)^{-1}\ dt$

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 ${A, B}$ are positive definite with ${A>B}$ in the sense of positive definite matrices (that is, ${A-B}$ is positive definite), then ${B^{-1} > A^{-1}}$. (Try to prove this using (6) instead!)

Exercise 6 If ${A}$ is an invertible ${n \times n}$ matrix and ${u, v}$ are ${n \times 1}$ vectors, establish the Sherman-Morrison formula

$\displaystyle (A + t uv^T)^{-1} = A^{-1} - \frac{t}{1 + t v^T A^{-1} u} A^{-1} uv^T A^{-1}$

whenever ${t}$ is a scalar such that ${1 + t v^T A^{-1} u}$ 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 ${F: {\bf C} \rightarrow {\bf C}}$ is an entire function, and ${\gamma}$ is a counterclockwise contour that goes around the spectrum of both ${H_0}$ and ${H_0+V}$, then we have

$\displaystyle F(H_0+V) = \frac{-1}{2\pi i} \int_\gamma F(z) R_V(z)\ dz$

and similarly

$\displaystyle F(H_0) = \frac{-1}{2\pi i} \int_\gamma F(z) R_0(z)\ dz$

and hence by (7) one has

$\displaystyle F(H_0+V) - F(H_0) = \frac{1}{2\pi i} \int_\gamma F(z) R_V(z) V F_0(z)\ dz;$

similarly, if ${H(t)}$ depends on ${t}$ in a continuously differentiable fashion, then

$\displaystyle \frac{d}{dt} F(H(t)) = \frac{1}{2\pi i} \int_\gamma F(z) (H(t) - zI)^{-1} H'(t) (z) (H(t)-zI)^{-1}\ dz$

as long as ${\gamma}$ goes around the spectrum of ${H(t)}$.

Exercise 7 If ${H(t)}$ is an ${n \times n}$ matrix depending continuously differentiably on ${t}$, and ${F: {\bf C} \rightarrow {\bf C}}$ is an entire function, establish the tracial chain rule

$\displaystyle \frac{d}{dt} \hbox{tr} F(H(t)) = \hbox{tr}(F'(H(t)) H'(t)).$

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

$\displaystyle \log A = \int_0^\infty (I + sI)^{-1} - (A + sI)^{-1}\ ds$

(with the constant term ${(I+tI)^{-1}}$ needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if ${A(t)}$ is a family of positive definite matrices depending continuously on ${t}$, that

$\displaystyle \frac{d}{dt} \log A(t) = \int_0^\infty (A(t) + sI)^{-1} A'(t) (A(t)+sI)^{-1}\ dt.$

This can be used for instance to show that ${\log}$ is a monotone increasing function, in the sense that ${\log A> \log B}$ whenever ${A > B > 0}$ in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference ${\log A - \log B}$ of the logarithm of two positive definite matrices ${A,B}$.

To compare the square root ${A^{1/2} - B^{1/2}}$ of two positive definite matrices ${A,B}$ 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

$\displaystyle A^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - I) t^{-1/2} \frac{dt}{t}$

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

$\displaystyle A^{1/2} - B^{1/2} = \frac{1}{\Gamma(-1/2)} \int_0^\infty (e^{-tA} - e^{-tB}) t^{-1/2} \frac{dt}{t}$

and one can use some of the previous identities to control ${e^{-tA} - e^{-tB}}$. This is pretty messy though. A third way to proceed is via implicit differentiation. If for instance ${A(t)}$ is a family of positive definite matrices depending continuously differentiably on ${t}$, we can differentiate the identity

$\displaystyle A(t)^{1/2} A(t)^{1/2} = A(t)$

to obtain

$\displaystyle A(t)^{1/2} \frac{d}{dt} A(t)^{1/2} + (\frac{d}{dt} A(t)^{1/2}) A(t)^{1/2} = \frac{d}{dt} A(t).$

This can for instance be solved using Exercise 5 to obtain

$\displaystyle \frac{d}{dt} A(t)^{1/2} = \int_0^\infty e^{-sA(t)^{1/2}} A'(t) e^{-sA(t)^{1/2}}\ ds$

and this can in turn be integrated to obtain a formula for ${A^{1/2} - B^{1/2}}$. 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: ${A > B > 0}$ implies ${A^{1/2} > B^{1/2} > 0}$.

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 ${(a_n)_{n=1}^\infty = (a_1, a_2, \dots)}$ of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit ${\lim_{n \rightarrow \infty} a_n}$, 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 ${L}$ such that for every ${\varepsilon>0}$, one has ${|a_n-L| \leq \varepsilon}$ for all sufficiently large ${n}$. 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 ${(a_n)_{n=1}^\infty}$ and ${(b_n)_{n=1}^\infty}$ are classically convergent sequences, then ${(a_n+b_n)_{n=1}^\infty}$ is also classically convergent with

$\displaystyle \lim_{n \rightarrow \infty} (a_n + b_n) = (\lim_{n \rightarrow \infty} a_n) + (\lim_{n \rightarrow \infty} b_n) \ \ \ \ \ (1)$

and similarly for any scalar ${c}$, ${(ca_n)_{n=1}^\infty}$ is classically convergent with

$\displaystyle \lim_{n \rightarrow \infty} (ca_n) = c \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (2)$

It is also an algebra homomorphism: ${(a_n b_n)_{n=1}^\infty}$ is also classically convergent with

$\displaystyle \lim_{n \rightarrow \infty} (a_n b_n) = (\lim_{n \rightarrow \infty} a_n) (\lim_{n \rightarrow \infty} b_n). \ \ \ \ \ (3)$

We also have shift invariance: if ${(a_n)_{n=1}^\infty}$ is classically convergent, then so is ${(a_{n+1})_{n=1}^\infty}$ with

$\displaystyle \lim_{n \rightarrow \infty} a_{n+1} = \lim_{n \rightarrow \infty} a_n \ \ \ \ \ (4)$

and more generally in fact for any injection ${\phi: {\bf N} \rightarrow {\bf N}}$, ${(a_{\phi(n)})_{n=1}^\infty}$ is classically convergent with

$\displaystyle \lim_{n \rightarrow \infty} a_{\phi(n)} = \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (5)$

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 ${(a_n)_{n=1}^\infty}$, one has

$\displaystyle \inf_n a_n \leq \lim_{n \rightarrow \infty} a_n \leq \sup_n a_n. \ \ \ \ \ (6)$

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 ${a_n = (-1)^n}$. On the one hand, one has ${a_n^2 = 1}$ for all ${n}$, so if one wishes to retain the homomorphism property (3), any “limit” of this sequence ${a_n}$ would have to necessarily square to ${1}$, that is to say it must equal ${+1}$ or ${-1}$. 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

$\displaystyle \limsup_{n \rightarrow \infty} a_n := \inf_N \sup_{n \geq N} a_n$

$\displaystyle \liminf_{n \rightarrow \infty} a_n := \sup_N \inf_{n \geq N} a_n$

which are well-defined real numbers for any bounded sequence ${(a_n)_{n=1}^\infty}$; 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

$\displaystyle \limsup_{n \rightarrow \infty} (a_n + b_n) \leq (\limsup_{n \rightarrow \infty} a_n) + (\limsup_{n \rightarrow \infty} b_n)$

for the limit superior, and the superadditivity property

$\displaystyle \liminf_{n \rightarrow \infty} (a_n + b_n) \geq (\liminf_{n \rightarrow \infty} a_n) + (\liminf_{n \rightarrow \infty} b_n)$

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

$\displaystyle \limsup_{n \rightarrow \infty} (-a_n) = - \liminf_{n \rightarrow \infty} a_n.$

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

$\displaystyle \mathrm{C}\!-\!\lim_{n \rightarrow \infty} a_n := \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n;$

if this limit exists, we say that the sequence is Césaro convergent. If the sequence ${(a_n)_{n=1}^\infty}$ 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 ${a_n= (-1)^n}$ discussed above is Césaro convergent, with a Césaro limit of ${0}$. However, there are still bounded sequences that do not have Césaro limit, such as ${a_n := \sin( \log n )}$ (exercise!). 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 ${\mathop{\widetilde \lim}_{n \rightarrow \infty} a_n}$, defined to be bounded linear functionals from the space ${\ell^\infty({\bf N})}$ of bounded real sequences to the real numbers ${{\bf R}}$ that extend the classical limit functional (defined on the space ${c_0({\bf N}) + {\bf R}}$ 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 ${(a_n)_{n=1}^\infty}$ and any number ${L}$ lying in the closed interval ${[\liminf_{n \rightarrow \infty} a_n, \limsup_{n \rightarrow \infty} a_n]}$, there exists at least one generalised limit functional ${\mathop{\widetilde \lim}_{n \rightarrow \infty}}$ that takes the value ${L}$ when applied to ${a_n}$; for instance, for any number ${\theta}$ in ${[-1,1]}$, there exists a generalised limit functional that assigns that number ${\theta}$ as the “limit” of the sequence ${a_n = (-1)^n}$. This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by ${(a_n)_{n=1}^\infty}$, 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 ${(a_n)_{n=1}^\infty}$ to classically converge to a limit ${L}$, namely that all generalised limits of this sequence must equal ${L}$.

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 presumably models of set theory without the axiom of choice in which no generalised limits exist, but I do not know of an explicit reference for this.

Generalised limits can obey the shift-invariance property (4) or the algebra homomorphism property (3), but as the above analysis of the sequence ${a_n = (-1)^n}$ 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 ${\mathop{\widetilde \lim}}$ is any generalised limit, then the Césaro-type functional ${(a_n)_{n=1}^\infty \mapsto \mathop{\widetilde \lim}_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n}$ 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 ${p\!-\!\lim_{n \rightarrow \infty}}$ that obeys (2), then for any subset ${A}$ of the natural numbers ${{\bf N}}$, the generalised limit ${p\!-\!\lim_{n \rightarrow \infty} 1_A(n)}$ must equal its own square (since ${1_A(n)^2 = 1_A(n)}$) and is thus either ${0}$ or ${1}$. If one defines ${p \subset 2^{2^{\bf N}}}$ to be the collection of all subsets ${A}$ of ${{\bf N}}$ for which ${p\!-\!\lim_{n \rightarrow \infty} 1_A(n) = 1}$, one can verify that ${p}$ obeys the axioms of a non-principal ultrafilter. Conversely, if ${p}$ is a non-principal ultrafilter, one can define the associated generalised limit ${p\!-\!\lim_{n \rightarrow \infty} a_n}$ of any bounded sequence ${(a_n)_{n=1}^\infty}$ to be the unique real number ${L}$ such that the sets ${\{ n \in {\bf N}: |a_n - L| \leq \varepsilon \}}$ lie in ${p}$ for all ${\varepsilon>0}$; one can check that this does indeed give a well-defined generalised limit that obeys (2). 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 ${a_n = (-1)^n}$ 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 ${L}$ if and only if all ultrafilter limits converge to ${L}$.

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.

We summarise 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 ${r_4(N)}$“, 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 ${N}$, define ${r_4(N)}$ to be the largest cardinality of a subset ${A}$ of ${[N] = \{1,\dots,N\}}$ which does not contain any non-trivial arithmetic progressions ${a, a+r, a+2r, a+3r}$ of length four (where “non-trivial” means that ${r}$ is non-zero). Trivially we have ${r_4(N) \leq N}$. In 1969, Szemerédi showed that ${r_4(N) = o(N)}$. 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 ${r_4(N) \ll N (\log \log N)^{-c}}$ for some absolute constant ${c>0}$. In the second paper in the above-mentioned series, we managed to improve this bound to ${r_4(N) \ll N \exp( - c \sqrt{\log \log N})}$. In this paper, we improve the bound further to ${r_4(N) \ll N (\log N)^{-c}}$, which seems to be the limit of the methods. (We remark that if we could take ${c}$ 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 ${r_3(N)}$ one can take any ${c}$ less than one.)

Most of the previous work on bounding ${r_4(N)}$ 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 ${A}$ of ${[N]}$ fails to contain arithmetic progressions of length four, one seeks to then locate a long subprogression of ${[N]}$ in which ${A}$ has increased density. This was the basic method for instance underlying our previous bound ${r_4(N) \ll N \exp( - c \sqrt{\log \log N})}$, as well as a finite field analogue of the bound ${r_4(N) \ll N (\log N)^{-c}}$; 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 ${A \subset [N]}$ has density ${\delta}$. Then one would expect a “randomly” selected arithmetic progression ${{\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r}}$ in ${[N]}$ (using the convention that random variables will be in boldface) to be contained in ${A}$ with probability about ${\delta^4}$. This is not true in general, however it was shown by Ben and myself that for any ${\eta>0}$, there was a set of shifts ${r \in [-N,N]}$ of cardinality ${\gg_{\delta,\eta} N}$, such that for any such ${r}$ one had

$\displaystyle {\bf P}( {\bf a}, {\bf a}+r, {\bf a}+2r, {\bf a}+3r \in A ) \geq \delta^4 - \eta$

if ${{\bf a}}$ was chosen uniformly at random from ${[N]}$. This easily implies that ${r_4(N) = o(N)}$, but does not give a particularly good bound on the decay rate, because the implied constant in the cardinality lower bound ${\gg_{\delta,\eta} N}$ is quite poor (in fact of tower-exponential type, due to the use of regularity lemmas!), and so one has to take ${N}$ to be extremely large compared to ${\delta,\eta}$ to avoid the possibility that the set of shifts in the above theorem consists only of the trivial shift ${r=0}$.

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 ${{\bf a}}$ and ${{\bf r}}$ of the length four progression. Namely, with ${A}$, ${\delta}$, ${\eta}$ as above, we are able to show that there exist random variables ${{\bf a}, {\bf r}}$, not necessarily independent, such that

$\displaystyle {\bf P}( {\bf a}, {\bf a}+{\bf r}, {\bf a}+2{\bf r}, {\bf a}+3{\bf r} \in A ) \geq \delta^4 - \eta \ \ \ \ \ (1)$

and such that we have the non-degeneracy bound

$\displaystyle {\bf P}( {\bf r} = 0 ) \ll \exp( - \eta^{-O(1)} ) / N.$

This then easily implies the main theorem.

The energy increment method is then deployed to locate a good pair ${({\bf a}, {\bf r})}$ 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 ${1_A}$ “behaves like” a globally quadratic function such as ${F( \alpha n^2 )}$, for some irrational ${\alpha}$ and some smooth periodic function ${F: {\bf R}/{\bf Z} \rightarrow {\bf R}}$ of mean ${\delta}$. If one then takes ${{\bf a}, {\bf r}}$ to be uniformly distributed in ${[N]}$ and ${[-\varepsilon N, \varepsilon N]}$ respectively for some small ${\varepsilon>0}$, with no coupling between the two variables, then the left-hand side of (1) is approximately of the form

$\displaystyle \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F(x) F(y) F(z) F(w) \ \ \ \ \ (2)$

where the integral is with respect to the probability Haar measure, and the constraint ${x-3y+3z-w=0}$ ultimately arises from the algebraic constraint

$\displaystyle \alpha {\bf a}^2 - 3 \alpha ({\bf a}+{\bf r})^2 + 3 \alpha ({\bf a}+2{\bf r})^2 - \alpha ({\bf a}+3{\bf r})^2 = 0.$

However, an application of the Cauchy-Schwarz inequality and Fubini’s theorem shows that the integral in (2) is at least ${(\int_{{\bf R}/{\bf Z}} F)^4}$, 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 ${[N]}$ is partitioned into some number of structured pieces ${B_c}$ (think of these as arithmetic progressions, or as “Bohr sets), and on each piece ${B_c}$, ${1_A}$ behaves like a locally quadratic function such as ${F_c( \alpha_c n^2 )}$, where ${\alpha_c}$ now varies with ${c}$, and the mean of ${F_c}$ will be approximately ${\delta}$ on the average after averaging in ${c}$ (weighted by the size of the pieces ${B_c}$). Now one should select ${{\bf a}}$ and ${{\bf r}}$ in the following coupled manner: first one chooses ${{\bf a}}$ uniformly from ${[N]}$, then one defines ${{\bf c}}$ to be the label ${c}$ such that ${{\bf a} \in B_c}$, and then selects ${{\bf r}}$ uniformly from a set ${B_{c,\varepsilon}}$ which is related to ${B_c}$ in much the same way that ${[-\varepsilon N, \varepsilon N]}$ is related to ${[N]}$. If one does this correctly, the analogue of (2) becomes

$\displaystyle {\bf E} \int_{(x,y,z,w) \in ({\bf R}/{\bf Z})^4: x-3y+3z-w = 0} F_{\mathbf c}(x) F_{\mathbf c}(y) F_{\mathbf c}(z) F_{\mathbf c}(w),$

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 ${1_A}$ which involves a decomposition of ${[N]}$ into structured pieces ${B_c}$, and a quadratic approximation to ${1_A}$ on each piece. If this approximation is accurate enough (or more precisely, if a certain (averaged) local Gowers uniformity norm ${U^3}$ of the error is small enough) to model the count in (1) (for random variables ${{\bf a}, {\bf r}}$ determined by the above partition of ${[N]}$ into pieces ${B_c}$), and if the frequencies (such as ${\alpha_c}$) 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 ${U^3}$. 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 ${1_A}$ in a manner that significantly increases its “energy” (basically an ${L^2}$ norm). Such energy increments cannot continue indefinitely, and when they terminate we obtain the desired claim.

There are existing inverse theorems for ${U^3}$ 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 “${1\%}$-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 “${99\%}$-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 ${99\%}$-structured homomorphism on pseudorandom subsets of Bohr sets to a ${100\%}$-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 ${1}$-form on ${{\bf R}^n}$ that varies linearly with the coordinates can be integrated to be the derivative of a quadratic function if we know that the ${1}$-form is closed). These portions of the paper are unfortunately rather technical, but broadly follow the methods already used in previous literature.