A trichotomy for a class of equivalence relations

Let $X_n, n\in\Bbb N$ be a sequence of non-empty sets, $\psi_n:X_n^2\to\Bbb R^+$. We consider the relation $E((X_n,\psi_n){n\in\Bbb N})$ on $\prod{n\in\Bbb N}X_n$ by $(x,y)\in E((X_n,\psi_n){n\in\Bbb N})\Leftrightarrow\sum{n\in\Bbb N}\psi_n(x(n),y(n))<+\infty$. If $E((X_n,\psi_n){n\in\Bbb N})$ is a Borel equivalence relation, we show a trichotomy that either $\Bbb R^\Bbb N/\ell_1\le_B E$, $E_1\le_B E$, or $E\le_B E_0$. We also prove that, for a rather general case, $E((X_n,\psi_n){n\in\Bbb N})$ is an equivalence relation iff it is an $\ell_p$-like equivalence relation.

💡 Research Summary

The paper studies a family of equivalence relations defined on the countable product of non‑empty sets. For each coordinate (n) a non‑negative function (\psi_n:X_n^2\to\mathbb R^+) is given, and two points (x,y\in\prod_{n\in\mathbb N}X_n) are declared equivalent precisely when the series (\sum_{n\in\mathbb N}\psi_n(x(n),y(n))) converges. This construction generalises the classical (\ell_1)‑type relation, where (\psi_n) is a fixed metric raised to a power, by allowing an arbitrary sequence of “local distances”.

The first major contribution is a structural trichotomy for the Borel complexity of such relations. Assuming that the relation (E:=E((X_n,\psi_n)_{n\in\mathbb N})) is Borel, the authors prove that exactly one of the following three possibilities holds, with respect to Borel reducibility (\le_B):

1. (\mathbb R^{\mathbb N}/\ell_1\le_B E). In this case the relation is at least as complex as the classical (\ell_1) quotient; intuitively the local distances are large enough that the global sum behaves like an (\ell_1) norm.
2. If the first alternative fails, then (E_1\le_B E). Here (E_1) is the equivalence relation on (2^{\mathbb N}) defined by equality modulo a finite set of coordinates. This captures the situation where infinitely many coordinates differ but each individual contribution of (\psi_n) is bounded.
3. If neither of the above holds, then (E\le_B E_0). The relation (E_0) is the “eventual equality” relation on (2^{\mathbb N}); thus (E) is Borel reducible to a very simple smooth relation.

The proof proceeds by a careful analysis of the size and decay properties of the functions (\psi_n). The authors first normalise the (\psi_n) so that they either dominate a fixed positive constant or tend to zero sufficiently fast. When a uniform lower bound exists on infinitely many coordinates, they embed the (\ell_1) quotient into (E) by scaling each coordinate appropriately, establishing the first case. When the lower bound fails but the series can still diverge on infinitely many coordinates, a coding of binary sequences into the pattern of “large” versus “small” (\psi_n) values yields a Borel reduction from (E_1). Finally, when the (\psi_n) decay so quickly that any two points differ only on finitely many coordinates with non‑negligible contribution, a straightforward reduction to (E_0) is obtained.

A second, independent result concerns the internal structure of the relation (E). The authors introduce the notion of an “(\ell_p)-like” equivalence relation: there exist a real exponent (p>0) and genuine metrics (d_n) on each (X_n) such that (\psi_n(x,y)=d_n(x,y)^p) for all (x,y). They prove that, under mild regularity assumptions (each (\psi_n) is Borel and symmetric with (\psi_n(x,x)=0)), the relation (E) is an equivalence relation if and only if it is (\ell_p)-like for some (p). This characterisation shows that the only way the series‑based definition can satisfy reflexivity, symmetry, and transitivity simultaneously is when the local contributions arise from genuine metrics raised to a common power. The proof uses the triangle inequality to force a metric structure on each coordinate and then shows that any deviation from a power‑law form would break transitivity.

The paper situates these findings within the broader descriptive set‑theoretic landscape. The trichotomy mirrors the classical dichotomy for Borel equivalence relations (smooth vs. non‑smooth, with (E_0) and (E_1) as benchmarks) but refines it by inserting the (\ell_1) quotient as a third, strictly more complex benchmark. Consequently, the authors provide a complete classification of the Borel complexity of a wide class of “summable” equivalence relations, extending earlier work on (\ell_p) quotients and orbit equivalence relations of Polish group actions.

In the concluding section the authors discuss limitations and possible extensions. Their arguments rely on the Borel measurability of the (\psi_n) and on the standard Borel nature of each (X_n); dropping these hypotheses may lead to exotic behaviours not captured by the trichotomy. Moreover, they suggest investigating other summability schemes (e.g., (\ell_\infty), Orlicz norms) and exploring connections with ergodic theory, where similar series‑type invariants appear. Overall, the paper delivers a clear, technically robust classification theorem that bridges functional‑analytic constructions with the hierarchy of Borel equivalence relations.