BOUNDING THE LOCAL DIMENSION OF THE CONVOLUTION OF MEASURES

The paper’s proof of Theorem 1.1 is correct and self-contained: it constructs a positive-measure subset A_ε of Nz with uniform control on μ(B(·,r)) at all sufficiently small r, then lower-bounds (μ*ν)(B(z,r)) and derives the desired upper local-dimension bound at z . By contrast, the model’s proof attempts to interchange limsup and sup over x (claiming limsup_{r→0} sup_{x∈E} a(x,r) ≤ sup_{x∈E} limsup_{r→0} a(x,r)), which is generally false without uniformity; the correct minimax inequality goes in the opposite direction. This invalid step is exactly what the paper avoids via the A_ε,n construction and Lemma 1.4 .