Lower basic scaling exponents versus lower outer Minkowski dimension

Determine whether an equality analogous to max_{i in {0,...,d-1}} m_i(A) = D_M(A) holds for the lower basic scaling exponents of a compact set A ⊂ R^d; specifically, ascertain whether the maximum of the lower i-th basic scaling exponents equals the lower outer Minkowski dimension, or provide a counterexample.

Background

The paper defines, for each i ∈ {0,…,d−1}, the i-th basic function β_i(t) derived from the i-th support measure μ_i(A;·) and introduces associated upper and lower basic scaling exponents m_i(A) and the lower analogs via liminf. Equation (4.4) (denoted in the text as \eqref{eq:dim_max_m}) shows that the maximum of the upper basic exponents equals the upper outer Minkowski dimension D_M(A).

The authors raise the question of whether an analogous equality could hold for the lower basic exponents. They conjecture it does not generally hold, making it an explicit unresolved question to either establish equality or construct counterexamples demonstrating failure.