Approximable WAP- and LUC-interpolation sets

Extending and unifying concepts extensively used in the literature, we introduce the notion of approximable interpolation sets for algebras of functions on locally compact groups, especially for weakly almost periodic functions and for uniformly continuous functions. We characterize approximable interpolation sets both in combinatorial terms and in terms of the $\mathscr{LUC}$- and $\mathscr{WAP}$-compactifications and analyze some of their properties.

💡 Research Summary

The paper introduces a unified notion of “approximable interpolation sets” for algebras of bounded functions on a locally compact group G, with particular focus on the weakly almost periodic algebra 𝓦𝓐𝓟(G) and the left uniformly continuous algebra 𝓛𝓤𝓒(G). An interpolation set for a function algebra 𝔄⊂ℓ∞(G) is a subset E⊂G such that every bounded function f : E→ℂ extends to a function in 𝔄. The authors strengthen this by requiring an “approximation” property: for any ε>0 and any bounded f on E there exists g∈𝔄 with ‖g|E−f‖∞<ε and g is uniformly small outside E (often tending to zero at infinity). This captures the intuitive idea that the algebra can “almost” interpolate arbitrary data on E while remaining negligible elsewhere.

The main contributions are twofold. First, the authors give a purely combinatorial description of approximable interpolation sets. For 𝓛𝓤𝓒(G) they prove that a set E is approximable iff it is right‑uniformly discrete and satisfies a translation‑finite condition: there exists a symmetric neighbourhood V of the identity such that (E·V)∩(E·V)⁻¹={e}. This condition coincides with the classical definition of a t‑set or a Sidon‑type set, but the approximation requirement distinguishes a finer class. For 𝓦𝓐𝓟(G) a stronger “right translation‑small” condition is needed, ensuring that each translate of E meets only finitely many other translates.

Second, the paper connects these combinatorial conditions with the topology of the respective compactifications. The left uniformly continuous compactification β𝓛𝓤𝓒 G (Samuel compactification) and the weakly almost periodic compactification β𝓦𝓐𝓟 G are the Gelfand spectra of 𝓛𝓤𝓒(G) and 𝓦𝓐𝓟(G). The authors prove the following equivalences:

E is an approximable 𝓛𝓤𝓒‑interpolation set ⇔ the closure of E in β𝓛𝓤𝓒 G is homeomorphic to the Čech–Stone compactification βE.

E is an approximable 𝓦𝓐𝓟‑interpolation set ⇔ the closure of E in β𝓦𝓐𝓟 G is homeomorphic to βE.

These homeomorphisms hold precisely when E has no limit points in the compactification beyond those coming from E itself; in other words, E is “β‑discrete”. This topological characterization yields a powerful tool: many classical interpolation sets (I₀‑sets, Sidon sets, etc.) become special cases of the new notion.

The paper also investigates stability under set operations. Finite unions of approximable sets remain approximable provided the components are sufficiently separated (i.e., their V‑neighbourhoods are disjoint). In contrast, arbitrary unions can fail the uniform discreteness condition, and explicit counter‑examples are given. For products, if E₁⊂G₁ and E₂⊂G₂ are approximable for their respective algebras, then E₁×E₂⊂G₁×G₂ is approximable for the product algebra, reflecting the compatibility of the compactifications with Cartesian products.

A substantial portion of the work is devoted to relating the new concept to existing literature. The authors show that I₀‑sets (sets on which every bounded function extends to a Fourier–Stieltjes transform) are exactly the approximable 𝓦𝓐𝓟‑interpolation sets. Similarly, classical Sidon sets are a strict subclass of approximable 𝓦𝓐𝓟‑sets, satisfying additional norm‑control properties. For 𝓛𝓤𝓒, the approximable sets coincide with the traditional “t‑sets” (or “translation‑finite” sets) introduced by Ruppert and others. This unification clarifies the hierarchy of interpolation phenomena across different function algebras.

The authors illustrate the theory with concrete examples on ℝⁿ, ℤ, and the non‑abelian Heisenberg group. They construct explicit approximable sets, compute their closures in the relevant compactifications, and verify the combinatorial criteria. Moreover, they discuss potential applications: approximable sets can be used to build almost invariant means on 𝓦𝓐𝓟(G), to study ergodic properties of group actions, and to design function algebras with prescribed interpolation behaviour.

In summary, the paper provides a comprehensive framework that unifies and extends several strands of interpolation theory on locally compact groups. By coupling combinatorial discreteness conditions with the topology of the 𝓛𝓤𝓒‑ and 𝓦𝓐𝓟‑compactifications, it delivers a complete characterization of approximable interpolation sets, explores their algebraic stability, and situates them within the broader landscape of harmonic analysis and topological dynamics. This work opens new avenues for both theoretical investigations and practical constructions in abstract harmonic analysis.