Monotonicity criteria for basic scaling exponents

Identify necessary and sufficient conditions under which the basic scaling exponents of a compact set A ⊂ R^d are monotone in the index, i.e., satisfy m_0(A) ≤ m_1(A) ≤ ··· ≤ m_{d−1}(A).

Background

The basic scaling exponents m_i(A) capture how components of the support measures contribute to the small-scale geometry of A. The examples show that these exponents need not be ordered (e.g., constructions where m_0(A) > m_1(A)).

Given such counterexamples to naive monotonicity, the authors pose the problem of characterizing precisely when monotonicity does hold, asking for necessary and sufficient geometric conditions on A.