Small doubling implies small tripling for balls of large radius

Small doubling implies small tripling for balls of large radius, Discrete Analysis 2025:9, 9 pp.

In additive combinatorics, the well-known Plünnecke–Ruzsa inequalities state that if \(A\) is a finite nonempty subset of an abelian group satisfying \(|2A| \le K|A|\) then \(|m A - n A| \le K^{m+n} |A|\) for all nonnegative integers \(m, n\). In particular it follows that the small-doubling hypothesis \(|2A| \le K|A|\) is sufficient to guarantee control of all iterated sum or difference sets.

In the case of nonabelian groups this has long been known not to hold. For example if \(A\) is the union of a finite subgroup \(H \le G\) and another element \(g\) then \(A^2 = Hg \cup gH \cup \{g^2\}\) has size comparable to \(A\), but \(A^3\) contains the subset \(HgH\), which may be as large as \(|H|^2\) if \(H \cap H^g = \{e\}\).

However, it does still follow from a version of the Ruzsa triangle inequality that the small-tripling condition \(|A^3| \le K|A|\) implies \(|A^{\epsilon_1} \cdots A^{\epsilon_m}| \le K^{3(m-2)} |A|\) for all \(\epsilon_i = \pm 1\), so it has become common in nonabelian additive combinatorics and the theory of growth in groups to assume a small-tripling condition.

It turns out that things are better if \(A\) is taken to be a ball \(B(r)\) (with respect to some generating set) for some large radius \(r\), which is a common and important special case.

In this case, a result of Breuillard and Tointon shows that the small-tripling condition can almost be weakened to a small-doubling condition. Precisely, if \(|B(2r+1)| \le K |B(r)|\) then \(|B(3r)| \le O_K(|B(r)|)\). This paper goes even further and shows that, if \(r\) is sufficiently large depending on \(K\), then in fact the true small-doubling hypothesis \(|B(2r)| \le K |B(r)|\) is indeed already sufficient to imply \(|B(3r)| \le O_K(1) |B(r)|\).

Variants of this result are also given for balls in locally compact groups and in locally finite vertex-transitive graphs.

Since there are several results in the literature assuming a small-tripling condition for balls, the results in this paper allow for the weakening of hypotheses in several previous results in the structure theory of groups and graphs, such as results of Easo and Hutchcroft on uniform finite presentation of groups of polynomial growth (see for instance their paper published recently in Discrete Analysis) and a quantitative version of the theorem of Trofimov on vertex-transitive graphs of polynomial growth.

The argument is short and clever, involving a combination of combinatorial and analytic techniques, including multiplicative energy and covering lemmas. The bounds obtained are effective and explicit.