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Strictness of the second inequality linking support and basic exponents

Demonstrate the existence of compact sets A ⊂ R^d for which the inequality that bounds the i-th upper support scaling exponent by the maximum of the upper basic scaling exponents over indices j ≤ i is strict; that is, show s_i(A) < max_{0 ≤ j ≤ i} m_j(A) for some i and A, or characterize when such strictness occurs.

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Background

Theorem “Properties of support scaling exponents” provides inequalities relating support scaling exponents s_i(A) to the maxima of basic scaling exponents m_j(A) over j ≤ i (displayed as \eqref{eq:s<max}). These relations come in two versions (corresponding to upper/lower or variation analogs).

The authors conjecture that the second of these two inequalities can be strict for specially designed sets, raising an explicit question about constructing such sets or describing conditions guaranteeing strictness.

References

Additionally, we conjecture that for certain specially designed sets, the second \leq-relatio n appearing in eq:s<max can also be strict.

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