Unequal contraction ratios for the absolute-continuity theorem

Determine whether the absolute-continuity and uniform density-integrability conclusions of Theorem 3.2 remain valid for equicontractive self-similar measures replaced by self-similar measures generated by an IFS with unequal contraction ratios, assuming dense rotations and finite $t$-energy for some $t>k$.

Background

Theorem 3.2 establishes absolute continuity of every kk-dimensional projection, together with uniform tail estimates for the densities, for equicontractive self-similar measures satisfying dense rotations and a finite-energy condition. The set theorem handles unequal contraction ratios through approximation by special subsystems, but this approximation does not yield the same result for a prescribed non-equicontractive measure. The obstruction is that the remaining smoothing scale depends on the word and is correlated with its rotation, suggesting the need for a joint renewal analysis of scale and direction.

References

Does Theorem~\ref{thm:energy} remain true for IFSs with different contraction ratios under the same dense-rotation and finite-energy assumptions?

— Absolute continuity and dimension conservation for self-similar sets and measures  (2609.29301 - Jin et al., 24 Sep 2026) in Section 7, “Prospects and open questions,” subsection “Unequal contraction ratios”

For an IFS $x\mapsto A_i x+a_i$ with invertible contracting linear parts, what assumptions on the generated linear group, together with finite $t$-energy for some $t>k$, imply absolute continuity of every $k$-dimensional projection, with uniform integrability of the densities? Under what further assumptions do the conditional measures have dimension $\mu-k$, and the set fibres have dimension $\dim A-k$ at almost every point of each projected image?

— Absolute continuity and dimension conservation for self-similar sets and measures  (2609.29301 - Jin et al., 24 Sep 2026) in Section 7, “Prospects and open questions,” subsection “Self-affine projections and dimension conservation”