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Typical equality in the first inequality relating support and basic exponents

Show that, for most (possibly all) compact sets A ⊂ R^d, the first inequality relating support and basic scaling exponents holds with equality; concretely, establish that s_i(A) = max_{0 ≤ j ≤ i} m_j(A) for the first of the two relations in the pair given in \eqref{eq:s<max}, or specify the exact scope of this equality.

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Background

The theorem relating support and basic exponents provides two inequalities bounding support exponents by maxima of basic exponents over lower indices (\eqref{eq:s<max}). While strictness is conjectured for the second inequality in some special constructions, the authors hypothesize that the first inequality is usually sharp.

This conjecture invites a rigorous proof of typical (or universal) equality, or a precise delineation of a class of sets for which equality always holds.

References

Conversely, we conjecture that the first \leq-relatio n in eq:s<max achieves equality for the majority of sets (possibly all) since no counterexamples have come to our attention.

eq:s:

$_i(A)\leq \max_{j\in\{0,\ldots, i\}}_j(A) \quad \text{ and } \quad _i(A)\leq \max_{j\in\{0,\ldots, i\}}_j(A). $

— Review of Steiner formulas in Fractal Geometry via Support measures and Complex Dimensions (2509.05227 - Radunović, 5 Sep 2025) in Section 7, Support Contents and the Parallel Set Perspective