Generalised lower Assouad-type dimensions and their interpolations

This paper investigates the analytic and structural properties of the $ϕ$-lower Assouad dimension, a generalized notion extending the lower Assouad dimension. We establish the equivalence of $ϕ$-lower Assouad dimensions with respect to the dimension functions, prove analytic properties related to the regularity of the $ϕ$-lower dimension, and analyse the role of rate windows in this context. Furthermore, we explore both positive and negative interpolation properties of the $ϕ$-lower dimension by presenting corresponding theorems that delineate these behaviors.

💡 Research Summary

The paper introduces a new family of dimensions, the ϕ‑lower Assouad dimensions, which generalize the classical lower Assouad dimension by allowing the small scale r to be prescribed as a function of the larger scale R via a “dimension function” ϕ. A dimension function ϕ:(0,1)→ℝ⁺ is required to be decreasing as R→0 and to satisfy R·ϕ(R)·log(1/R)→∞, ensuring that the exponent R^{−ϕ(R)} grows rapidly enough for the analysis. The ϕ‑lower dimension of a bounded doubling metric space F is defined by
 dim_ϕ L F = sup{ s≥0 : ∃C>0 such that ∀0<r=R^{1+ϕ(R)}<R<1, inf_{x∈F} N_r(B(x,R)∩F) ≥ C R^{−ϕ(R)s} }.
A quasi‑ϕ version replaces the right‑hand side by C(R/r)^s, and a modified version takes the supremum over all subsets E⊂F. The authors first establish an equivalence theorem (Theorem 1): if two dimension functions ϕ and ψ satisfy lim_{R→0}ϕ(R)/ψ(R)=1, then for every bounded set F we have dim_ϕ L F = dim_ψ L F. This shows that the precise asymptotic shape of ϕ does not affect the resulting dimension as long as the two functions are asymptotically equivalent.

The paper then studies the “rate windows” W_ϕ = {ϕ_α : ϕ_α(R)=ϕ(R)/α, α>0}. Theorem 2 proves that the ϕ‑lower dimension can be expressed as an infimum over this family: dim_ϕ L F = inf_{0<α<1} dim_{ϕ_α} L F. Consequently, varying α yields a continuous spectrum of dimensions, analogous to the lower Assouad spectrum but now governed by the chosen ϕ.

The central contribution concerns interpolation between the classical lower Assouad dimension dim L F and the quasi‑lower Assouad dimension dim q L F. Theorem 3 shows that for any target value s lying between these two extremes, one can construct a dimension function ϕ such that both the ϕ‑lower and quasi‑ϕ‑lower dimensions of F equal s. In other words, the ϕ‑framework can fill any gap in the interval