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I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that

\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)

whenever {A,B} are compact non-empty subsets of a compact connected additive group {G} with probability Haar measure {\mu}.  (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)

The connectedness of {G} is essential, otherwise one could form counterexamples involving proper subgroups of {G} of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B) \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)

which in turn easily follows from the identity {(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)} and the inclusion {(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking {G} to be the unit circle {G = {\bf R}/{\bf Z}} and {A,B} to be arcs in that circle (obeying (2)). A bit more generally, if {G} is an arbitrary connected compact abelian group and {\xi: G \rightarrow {\bf R}/{\bf Z}} is a non-trivial character (i.e., a continuous homomorphism), then {\xi} must be surjective (as {{\bf R}/{\bf Z}} has no non-trivial connected subgroups), and one can take {A = \xi^{-1}(I)} and {B = \xi^{-1}(J)} for some arcs {I,J} in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then {A,B} must be close (in measure) to an example of the above form {A = \xi^{-1}(I), B = \xi^{-1}(J)}. Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset {A+B} is replaced by the partial sumset {A +_\varepsilon B} consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call {(A,B)} a critical pair if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in {A,B}, and invariant with respect to translation of either {A} or {B}. Furthermore, from the submodularity inequality (3), one can show that if {(A_1,B)} and {(A_2,B)} are critical pairs (with {\mu(A_1 \cap A_2)} and {1 - \mu(A_1 \cup A_2) - \mu(B)} positive), then {(A_1 \cap A_2,B)} and {(A_1 \cup A_2, B)} are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when {A,B} come from arcs of a circle.) Similarly, from associativity {(A+B)+C = A+(B+C)}, one can show that if {(A,B)} and {(A+B,C)} are critical pairs, then so are {(B,C)} and {(A,B+C)}.

One can combine these closure properties to obtain further ones. For instance, suppose {A,B} is such that {\mu(A+B) < 2 \mu(A)}, so that all translates {A+b}, {b \in B} intersect each other in a set of positive measure. Suppose also that {(A,C)} is a critical pair and {1-\mu(A+B) - \mu(C) > 0}. Then (cheating a little bit), one can show that {(A+B,C)} is also a critical pair, basically because {A+B} is the union of the {A+b}, {b \in B}, the {(A+b,C)} are all critical pairs, and the {A+b} all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate {A+B} by the union of finitely many of the {A+b}, then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair {(A,B)} and end up with a small set {C} such that {(A,C)} and {(kC,C)} are also critical pairs for all {1 \leq k \leq 10^4} (say), where {kC} is the {k}-fold sumset of {C}. (Intuitively, if {A,B} are thought of as secretly coming from the pullback of arcs {I,J} by some character {\xi}, then {C} should be the pullback of a much shorter arc by the same character.) In particular, {C} exhibits linear growth, in that {\mu(kC) = k\mu(C)} for all {1 \leq k \leq 10^4}. One can now use standard technology from inverse sumset theory to show first that {C} has a very large Fourier coefficient (and thus is biased with respect to some character {\xi}), and secondly that {C} is in fact almost of the form {C = \xi^{-1}(K)} for some arc {K}, from which it is not difficult to conclude similar statements for {A} and {B} and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

 

[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman.  Thanks to John Griesmer for pointing out the error.]

A basic object of study in multiplicative number theory are the arithmetic functions: functions {f: {\bf N} \rightarrow {\bf C}} from the natural numbers to the complex numbers. Some fundamental examples of such functions include

Given an arithmetic function {f}, we are often interested in statistics such as the summatory function

\displaystyle \sum_{n \leq x} f(n), \ \ \ \ \ (1)

 

the logarithmically (or harmonically) weighted summatory function

\displaystyle \sum_{n \leq x} \frac{f(n)}{n}, \ \ \ \ \ (2)

 

or the Dirichlet series

\displaystyle {\mathcal D}[f](s) := \sum_n \frac{f(n)}{n^s}.

In the latter case, one typically has to first restrict {s} to those complex numbers whose real part is large enough in order to ensure the series on the right converges; but in many important cases, one can then extend the Dirichlet series to almost all of the complex plane by analytic continuation. One is also interested in correlations involving additive shifts, such as {\sum_{n \leq x} f(n) f(n+h)}, but these are significantly more difficult to study and cannot be easily estimated by the methods of classical multiplicative number theory.

A key operation on arithmetic functions is that of Dirichlet convolution, which when given two arithmetic functions {f,g: {\bf N} \rightarrow {\bf C}}, forms a new arithmetic function {f*g: {\bf N} \rightarrow {\bf C}}, defined by the formula

\displaystyle f*g(n) := \sum_{d|n} f(d) g(\frac{n}{d}).

Thus for instance {1*1 = d_2}, {1 * \Lambda = L}, {1 * \mu = \delta}, and {\delta * f = f} for any arithmetic function {f}. Dirichlet convolution and Dirichlet series are related by the fundamental formula

\displaystyle {\mathcal D}[f * g](s) = {\mathcal D}[f](s) {\mathcal D}[g](s), \ \ \ \ \ (3)

 

at least when the real part of {s} is large enough that all sums involved become absolutely convergent (but in practice one can use analytic continuation to extend this identity to most of the complex plane). There is also the identity

\displaystyle {\mathcal D}[Lf](s) = - \frac{d}{ds} {\mathcal D}[f](s), \ \ \ \ \ (4)

 

at least when the real part of {s} is large enough to justify interchange of differentiation and summation. As a consequence, many Dirichlet series can be expressed in terms of the Riemann zeta function {\zeta = {\mathcal D}[1]}, thus for instance

\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s)

\displaystyle {\mathcal D}[L](s) = - \zeta'(s)

\displaystyle {\mathcal D}[\delta](s) = 1

\displaystyle {\mathcal D}[\mu](s) = \frac{1}{\zeta(s)}

\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)}.

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers {{\bf N}}, which form a rather complicated abelian semigroup with respect to multiplication (in particular the set of generators is the set of prime numbers). One can obtain a simpler analogue of the subject by working instead with the half-infinite interval {{\bf N}_\infty := [1,+\infty)}, which is a much simpler abelian semigroup under multiplication (being a one-dimensional Lie semigroup). (I will think of this as a sort of “completion” of {{\bf N}} at the infinite place {\infty}, hence the terminology.) Accordingly, let us define a continuous arithmetic function to be a locally integrable function {f: {\bf N}_\infty \rightarrow {\bf C}}. The analogue of the summatory function (1) is then an integral

\displaystyle \int_1^x f(t)\ dt,

and similarly the analogue of (2) is

\displaystyle \int_1^x \frac{f(t)}{t}\ dt.

The analogue of the Dirichlet series is the Mellin-type transform

\displaystyle {\mathcal D}_\infty[f](s) := \int_1^\infty \frac{f(t)}{t^s}\ dt,

which will be well-defined at least if the real part of {s} is large enough and if the continuous arithmetic function {f: {\bf N}_\infty \rightarrow {\bf C}} does not grow too quickly, and hopefully will also be defined elsewhere in the complex plane by analytic continuation.

For instance, the continuous analogue of the discrete constant function {1: {\bf N} \rightarrow {\bf C}} would be the constant function {1_\infty: {\bf N}_\infty \rightarrow {\bf C}}, which maps any {t \in [1,+\infty)} to {1}, and which we will denote by {1_\infty} in order to keep it distinct from {1}. The two functions {1_\infty} and {1} have approximately similar statistics; for instance one has

\displaystyle \sum_{n \leq x} 1 = \lfloor x \rfloor \approx x-1 = \int_1^x 1\ dt

and

\displaystyle \sum_{n \leq x} \frac{1}{n} = H_{\lfloor x \rfloor} \approx \log x = \int_1^x \frac{1}{t}\ dt

where {H_n} is the {n^{th}} harmonic number, and we are deliberately vague as to what the symbol {\approx} means. Continuing this analogy, we would expect

\displaystyle {\mathcal D}[1](s) = \zeta(s) \approx \frac{1}{s-1} = {\mathcal D}_\infty[1_\infty](s)

which reflects the fact that {\zeta} has a simple pole at {s=1} with residue {1}, and no other poles. Note that the identity {{\mathcal D}_\infty[1_\infty](s) = \frac{1}{s-1}} is initially only valid in the region {\mathrm{Re} s > 1}, but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at {1}, and so one can define {{\mathcal D}_\infty[1_\infty]} in this region also.

In a similar vein, the logarithm function {L: {\bf N} \rightarrow {\bf C}} is approximately similar to the logarithm function {L_\infty: {\bf N}_\infty \rightarrow {\bf C}}, giving for instance the crude form

\displaystyle \sum_{n \leq x} L(n) = \log \lfloor x \rfloor! \approx x \log x - x = \int_1^\infty L_\infty(t)\ dt

of Stirling’s formula, or the Dirichlet series approximation

\displaystyle {\mathcal D}[L](s) = -\zeta'(s) \approx \frac{1}{(s-1)^2} = {\mathcal D}_\infty[L_\infty](s).

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure {\frac{dt}{t}}: given two continuous arithmetic functions {f_\infty, g_\infty: {\bf N}_\infty \rightarrow {\bf C}}, one can define their convolution {f_\infty *_\infty g_\infty: {\bf N}_\infty \rightarrow {\bf C}} by the formula

\displaystyle f_\infty *_\infty g_\infty(t) := \int_1^t f_\infty(s) g_\infty(\frac{t}{s}) \frac{ds}{s}.

Thus for instance {1_\infty * 1_\infty = L_\infty}. A short computation using Fubini’s theorem shows the analogue

\displaystyle D_\infty[f_\infty *_\infty g_\infty](s) = D_\infty[f_\infty](s) D_\infty[g_\infty](s)

of (3) whenever the real part of {s} is large enough that Fubini’s theorem can be justified; similarly, differentiation under the integral sign shows that

\displaystyle D_\infty[L_\infty f_\infty](s) = -\frac{d}{ds} D_\infty[f_\infty](s) \ \ \ \ \ (5)

 

again assuming that the real part of {s} is large enough that differentiation under the integral sign (or some other tool like this, such as the Cauchy integral formula for derivatives) can be justified.

Direct calculation shows that for any complex number {\rho}, one has

\displaystyle \frac{1}{s-\rho} = D_\infty[ t \mapsto t^{\rho-1} ](s)

(at least for the real part of {s} large enough), and hence by several applications of (5)

\displaystyle \frac{1}{(s-\rho)^k} = D_\infty[ t \mapsto \frac{1}{(k-1)!} t^{\rho-1} \log^{k-1} t ](s)

for any natural number {k}. This can lead to the following heuristic: if a Dirichlet series {D[f](s)} behaves like a linear combination of poles {\frac{1}{(s-\rho)^k}}, in that

\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{(s-\rho)^{k_\rho}}

for some set {\rho} of poles and some coefficients {c_\rho} and natural numbers {k_\rho} (where we again are vague as to what {\approx} means, and how to interpret the sum {\sum_\rho} if the set of poles is infinite), then one should expect the arithmetic function {f} to behave like the continuous arithmetic function

\displaystyle t \mapsto \sum_\rho \frac{c_\rho}{(k_\rho-1)!} t^{\rho-1} \log^{k_\rho-1} t.

In particular, if we only have simple poles,

\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{s-\rho}

then we expect to have {f} behave like continuous arithmetic function

\displaystyle t \mapsto \sum_\rho c_\rho t^{\rho-1}.

Integrating this from {1} to {x}, this heuristically suggests an approximation

\displaystyle \sum_{n \leq x} f(n) \approx \sum_\rho c_\rho \frac{x^\rho-1}{\rho}

for the summatory function, and similarly

\displaystyle \sum_{n \leq x} \frac{f(n)}{n} \approx \sum_\rho c_\rho \frac{x^{\rho-1}-1}{\rho-1},

with the convention that {\frac{x^\rho-1}{\rho}} is {\log x} when {\rho=0}, and similarly {\frac{x^{\rho-1}-1}{\rho-1}} is {\log x} when {\rho=1}. One can make these sorts of approximations more rigorous by means of Perron’s formula (or one of its variants) combined with the residue theorem, provided that one has good enough control on the relevant Dirichlet series, but we will not pursue these rigorous calculations here. (But see for instance this previous blog post for some examples.)

For instance, using the more refined approximation

\displaystyle \zeta(s) \approx \frac{1}{s-1} + \gamma

to the zeta function near {s=1}, we have

\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s) \approx \frac{1}{(s-1)^2} + \frac{2 \gamma}{s-1}

we would expect that

\displaystyle d_2 \approx L_\infty + 2 \gamma

and thus for instance

\displaystyle \sum_{n \leq x} d_2(n) \approx x \log x - x + 2 \gamma x

which matches what one actually gets from the Dirichlet hyperbola method (see e.g. equation (44) of this previous post).

Or, noting that {\zeta(s)} has a simple pole at {s=1} and assuming simple zeroes elsewhere, the log derivative {-\zeta'(s)/\zeta(s)} will have simple poles of residue {+1} at {s=1} and {-1} at all the zeroes, leading to the heuristic

\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)} \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho}

suggesting that {\Lambda} should behave like the continuous arithmetic function

\displaystyle t \mapsto 1 - \sum_\rho t^{\rho-1}

leading for instance to the summatory approximation

\displaystyle \sum_{n \leq x} \Lambda(n) \approx x - \sum_\rho \frac{x^\rho-1}{\rho}

which is a heuristic form of the Riemann-von Mangoldt explicit formula (see Exercise 45 of these notes for a rigorous version of this formula).

Exercise 1 Go through some of the other explicit formulae listed at this Wikipedia page and give heuristic justifications for them (up to some lower order terms) by similar calculations to those given above.

Given the “adelic” perspective on number theory, I wonder if there are also {p}-adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as {\sum_{n \leq x: n = a \hbox{ mod } p^j} f(n)}. A key problem here is that there does not seem to be any good interpretation of the expression {\frac{1}{t^s}} when {s} is complex and {t} is a {p}-adic number, so it is not clear that one can analyse a Dirichlet series {p}-adically. For similar reasons, we don’t have a canonical way to define {\chi(t)} for a Dirichlet character {\chi} (unless its conductor happens to be a power of {p}), so there doesn’t seem to be much to say in the {q}-aspect either.

Alice Guionnet, Assaf Naor, Gilles Pisier, Sorin Popa, Dimitri Shylakhtenko, and I are organising a three month program here at the Institute for Pure and Applied Mathematics (IPAM) on the topic of Quantitative Linear Algebra.  The purpose of this program is to bring together mathematicians and computer scientists (both junior and senior) working in various quantitative aspects of linear operators, particularly in large finite dimension.  Such aspects include, but are not restricted to discrepancy theory, spectral graph theory, random matrices, geometric group theory, ergodic theory, von Neumann algebras, as well as specific research directions such as the Kadison-Singer problem, the Connes embedding conjecture and the Grothendieck inequality.  There will be several workshops and tutorials during the program (for instance I will be giving a series of introductory lectures on random matrix theory).

While we already have several confirmed participants, we are still accepting applications for this program until Dec 4; details of the application process may be found at this page.

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