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In 1964, Kemperman established the following result:
Remark 1 The estimate is sharp, as can be seen by considering the case when is a unit circle, and are arcs; similarly if is any compact connected group that projects onto the circle. The connectedness hypothesis is essential, as can be seen by considering what happens if and are a non-trivial open subgroup of . For locally compact connected groups which are unimodular but not compact, there is an analogous statement, but with now a Haar measure instead of a Haar probability measure, and the right-hand side replaced simply by . The case when is a torus is due to Macbeath, and the case when is a circle is due to Raikov. The theorem is closely related to the Cauchy-Davenport inequality; indeed, it is not difficult to use that inequality to establish the circle case, and the circle case can be used to deduce the torus case by considering increasingly dense circle subgroups of the torus (alternatively, one can also use Kneser’s theorem).
By inner regularity, the hypothesis that are compact can be replaced with Borel measurability, so long as one then adds the additional hypothesis that is also Borel measurable.
A short proof of Kemperman’s theorem was given by Ruzsa. In this post I wanted to record how this argument can be used to establish the following more “robust” version of Kemperman’s theorem, which not only lower bounds , but gives many elements of some multiplicity:
Indeed, Theorem 1 can be deduced from Theorem 2 by dividing (1) by and then taking limits as . The bound in (1) is sharp, as can again be seen by considering the case when are arcs in a circle. The analogous claim for cyclic groups for prime order was established by Pollard, and for general abelian groups by Green and Ruzsa.
for any compact set . Our task is to establish that whenever .
We first verify the extreme cases. If , then , and so in this case (since ). At the other extreme, if , then from the inclusion-exclusion principle we see that , and so again in this case.
and thus (noting that the quantities on the left are closer to each other than the quantities on the right)
at which point (2) follows by integrating over and then using the inclusion-exclusion principle.
Now introduce the function
for . From the preceding discussion vanishes at the endpoints ; our task is to show that is non-negative in the interior region . Suppose for contradiction that this was not the case. It is easy to see that is continuous (indeed, it is even Lipschitz continuous), so there must be at which is a local minimum and not locally constant. In particular, . But for any with , we have the translation-invariance
for any , and hence by (2)
Note that depends continuously on , equals when is the identity, and has an average value of . As is connected, we thus see from the intermediate value theorem that for any , we can find such that , and thus by inclusion-exclusion . By definition of , we thus have
Taking infima in (and noting that the hypotheses on are independent of ) we conclude that
for all . As is a local minimum and is arbitrarily small, this implies that is locally constant, a contradiction. This establishes Theorem 2.
We observe the following corollary:
Corollary 3 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of , and let . Then one has the pointwise estimate
if , and
Once again, the bounds are completely sharp, as can be seen by computing when are arcs of a circle. For quasirandom , one can do much better than these bounds, as discussed in this recent blog post; thus, the abelian case is morally the worst case here, although it seems difficult to convert this intuition into a rigorous reduction.
Proof: By cyclic permutation we may take . For any
we can bound
where we used Theorem 2 to obtain the third line. Optimising in , we obtain the claim.
In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with the minimal assumption of local compactness on a topological group , and seeing what kind of structures one can build using this assumption. (For simplicity we shall mostly confine our discussion to global groups rather than local groups for now.) In view of the preceding notes, we would like to see two types of structures emerge in particular:
- representations of into some more structured group, such as a matrix group ; and
- metrics on that capture the escape and commutator structure of (i.e. Gleason metrics).
To build either of these structures, a fundamentally useful tool is that of (left-) Haar measure – a left-invariant Radon measure on . (One can of course also consider right-Haar measures; in many cases (such as for compact or abelian groups), the two concepts are the same, but this is not always the case.) This concept generalises the concept of Lebesgue measure on Euclidean spaces , which is of course fundamental in analysis on those spaces.
Haar measures will help us build useful representations and useful metrics on locally compact groups . For instance, a Haar measure gives rise to the regular representation that maps each element of to the unitary translation operator on the Hilbert space of square-integrable measurable functions on with respect to this Haar measure by the formula
(The presence of the inverse is convenient in order to obtain the homomorphism property without a reversal in the group multiplication.) In general, this is an infinite-dimensional representation; but in many cases (and in particular, in the case when is compact) we can decompose this representation into a useful collection of finite-dimensional representations, leading to the Peter-Weyl theorem, which is a fundamental tool for understanding the structure of compact groups. This theorem is particularly simple in the compact abelian case, where it turns out that the representations can be decomposed into one-dimensional representations , better known as characters, leading to the theory of Fourier analysis on general compact abelian groups. With this and some additional (largely combinatorial) arguments, we will also be able to obtain satisfactory structural control on locally compact abelian groups as well.
The link between Haar measure and useful metrics on is a little more complicated. Firstly, once one has the regular representation , and given a suitable “test” function , one can then embed into (or into other function spaces on , such as or ) by mapping a group element to the translate of in that function space. (This map might not actually be an embedding if enjoys a non-trivial translation symmetry , but let us ignore this possibility for now.) One can then pull the metric structure on the function space back to a metric on , for instance defining an -based metric
if is continuous and compactly supported (with denoting the supremum norm). These metrics tend to have several nice properties (for instance, they are automatically left-invariant), particularly if the test function is chosen to be sufficiently “smooth”. For instance, if we introduce the differentiation (or more precisely, finite difference) operators
(so that ) and use the metric (1), then a short computation (relying on the translation-invariance of the norm) shows that
for all . This suggests that commutator estimates, such as those appearing in the definition of a Gleason metric in Notes 2, might be available if one can control “second derivatives” of ; informally, we would like our test functions to have a “” type regularity.
If was already a Lie group (or something similar, such as a local group) then it would not be too difficult to concoct such a function by using local coordinates. But of course the whole point of Hilbert’s fifth problem is to do without such regularity hypotheses, and so we need to build test functions by other means. And here is where the Haar measure comes in: it provides the fundamental tool of convolution
between two suitable functions , which can be used to build smoother functions out of rougher ones. For instance:
Exercise 1 Let be continuous, compactly supported functions which are Lipschitz continuous. Show that the convolution using Lebesgue measure on obeys the -type commutator estimate
for all and some finite quantity depending only on .
This exercise suggests a strategy to build Gleason metrics by convolving together some “Lipschitz” test functions and then using the resulting convolution as a test function to define a metric. This strategy may seem somewhat circular because one needs a notion of metric in order to define Lipschitz continuity in the first place, but it turns out that the properties required on that metric are weaker than those that the Gleason metric will satisfy, and so one will be able to break the circularity by using a “bootstrap” or “induction” argument.
We will discuss this strategy – which is due to Gleason, and is fundamental to all currently known solutions to Hilbert’s fifth problem – in later posts. In this post, we will construct Haar measure on general locally compact groups, and then establish the Peter-Weyl theorem, which in turn can be used to obtain a reasonably satisfactory structural classification of both compact groups and locally compact abelian groups.