Approximate lattices and Meyer sets in nilpotent Lie groups

Approximate lattices and Meyer sets in nilpotent Lie groups, Discrete Analysis 2020:1, 18 pp.

A central result in additive combinatorics, Freiman’s theorem, describes the structure of any finite set $A$ of integers with the property that its sumset $A+A$ has size at most $C|A|$. Freiman’s theorem, and also the ingredients of a second proof of the theorem discovered by Imre Ruzsa, have many applications, and in due course it became apparent that a suitable non-Abelian version of the theorem would also have many applications.

There are examples that show that if $A$ is a finite subset of a general group $G$, then the hypothesis that $A.A$ is small is not strong enough to yield the kind of structural information one seeks, but that can be dealt with by adopting the stronger condition that $A.A.A$ is small. However, a more convenient hypothesis that is essentially equivalent to this “small tripling” is that there should exist a small subset $F\subset G$ such that $A.A\subset F.A$. If $F$ has cardinality $K$, then such a set is called a $K$-approximate group. A non-Abelian version of Freiman’s theorem that describes the structure of approximate groups was formulated and proved in a seminal paper of Breuillard, Green and Tao, and their result did indeed have a number of important consequences.

A noticeable feature of the definition of a $K$-approximate group is that, unlike the condition that $A.A.A$ should be small, it applies just as well if the set $A$ is infinite. It is tempting, therefore, to ask what can be said about infinite approximate groups. It is to that general question that this paper makes a contribution.

It turns out that infinite approximate groups are too diverse for there to be much hope of proving a structure theorem for them, so in order to bring them under control people have imposed additional conditions. In particular Michael Björklund and Tobias Hartnick introduced the notion of a uniform approximate lattice. This is a subset $\Lambda$ of a locally compact group $G$ with the following three properties.

(i) $\Lambda$ is an approximate group.

(ii) There exists a compact set $K$ such that $K.\Lambda=G$.

(iii) There exists a compact neighbourhood $K'$ of the identity such that no translate of $K'$ contains more than one point of $\Lambda$.

If $\Lambda$ is a subgroup of $\mathbb R^d$, then the second condition tells us that $\Lambda$ does not live in a proper subspace and the third tells us that it is a discrete set, so it is a lattice in the usual sense. In the general case, we also call $\Lambda$ a lattice if it is a subgroup of $G$.

Approximate lattices in Abelian groups were studied (under a different name) by Yves Meyer, who gave a complete characterization. An interesting example is the set of all sums $\sum_{i=1}^n\epsilon_i\phi^i$, where $\phi$ is the golden ratio and the coefficients $\epsilon_i$ come from the set $\{-1,0,1\}$. This exploits the fact that $1+\phi=\phi^2$ and also the fact that the conjugate of $\phi$ (which happens to equal $-\phi^{-1}$) has modulus less than 1, making $\phi$ a Pisot number.

More generally, one can create interesting examples of approximate lattices using what is called a cut and project scheme, an idea that goes back to Meyer’s work, and which was later used to create aperiodic tilings. To create an approximate lattice in $G$, one takes another group $H$ (also locally compact) and a lattice $\Gamma$ in $G\times H$ that satisfies suitable conditions. One then “cuts out a strip” of $G\times H$, namely a set $S=G\times W_0$, where $W_0$ is a compact neighbourhood of the identity of $H$, and then projects down to $G$ all the points of $\Gamma$ that lie in $S$.

Meyer showed that if $G$ is Abelian, then all approximate lattices are built in essentially this way. (For a precise statement of the theorem, see Section 2 of the paper, where it appears as Theorem 2.10.) Björklund and Hartnick asked whether this result could be extended to all second countable locally compact groups.

This paper gives a positive answer in the case of connected nilpotent Lie groups. Not all such groups contain approximate lattices, and another result of the paper is to characterize the ones that do. The condition is that the underlying vector space of the associated Lie algebra should have a basis with the property that the product of any two basis vectors, when expanded in terms of the basis, should have coefficients that are real and algebraic.

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