SLICES AND DISTANCES: ON TWO PROBLEMS OF FURSTENBERG AND FALCONER

The paper states and proves that for homogeneous digit IFS measures µ_X, µ_Y with log a/log b irrational, one has D(µ_X * S_u µ_Y, ∞) = min(α+β, 1) for all u ≠ 0 (Theorem 3.2), via a dynamical self-similarity framework, a subadditive cocycle, unique ergodicity, exponential separation, and an inverse theorem for L^q norms, then passing q → ∞; see the statement and strategy around Theorem 3.2 and Sections 3.3–3.7 . The candidate solution instead cites a convolution theorem for L^q-dimensions of homogeneous self-similar measures (Shmerkin 2019) to obtain D(µ_X * S_u µ_Y, q) = min(α+β, 1) for all q>1, and then uses D(·,∞)=lim_{q→∞} D(·,q) (as defined in the paper) to conclude the same equality for q=∞; see the paper’s definition/equality D(µ,∞)=lim_{q→∞} D(µ,q) and Frostman-exponent characterization in §3.3 . Both approaches reach the same conclusion; the paper’s proof is dynamical and uses an inverse theorem at finite q (not directly at q=∞, which the paper notes is delicate ), whereas the model leverages an external theorem on L^q convolution dimensions and then passes to the limit. Minor issues: the model informally asserts strong separation for digit IFS (endpoints may touch, but the needed exponential separation and L^q computations still go through), and it compresses the dynamical ingredients the paper makes explicit. Net: both are correct; the model’s route is a citation-driven shortcut as opposed to the paper’s detailed (but compatible) argument.