THE FURSTENBERG SET AND ITS RANDOM VERSION

The paper proves both target statements: dim_H W(S) ≥ 0.451621 via a 6-adic Moran construction and asymptotics of the Weyl sums, and that W(S) supports a Rajchman measure via a carefully designed Bernoulli–6-adic construction (Theorems 2.6 and 2.8). The candidate solution also establishes the two conclusions by a different route: it builds a Cantor set of points admitting very strong rational approximations with denominators in S to force a persistent bias in exponential sums, and sketches a Rajchman measure via random recursive (Bluhm-type) constructions. However, the model’s dimension accounting contains a factor-of-two slip and a minor coefficient mismatch; its Rajchman measure argument relies on external results and omits some verification details. Overall, both deliver the results; the paper’s proof is complete and stronger quantitatively, while the model’s is a weaker but essentially correct alternative.