Estimates on the Dimension of Self-Similar Measures with Overlaps

The paper (Feng–Feng, arXiv:2103.01700, v1 dated 2021-03-02) gives a computer-assisted proof establishing a uniform lower bound dim(μβ) ≥ 0.98040856 for all β ∈ (1,2), and further shows that dim(μβ) > dim(μβ3) for all β ∈ (√2,2) outside the 10^{-8}-neighborhood of the tribonacci number β3 (the largest root of x^3−x^2−x−1) via an explicit parameter sweep and rigorously controlled estimates; see the abstract and Theorem 1.2 as well as the proof details and computation tables (including the smallest interval achieving the minimum bound) in Section 5 and Table 6 . The approach is based on a general lower-bounding inequality for self-similar measures (Theorem 1.1) derived from projection entropy, combined with carefully designed partitions and certified upper bounds on the relevant preimage measures, yielding uniform bounds over 132,530 subintervals that together cover the parameter range . The model’s claim that such results were likely open as of the cutoff is incorrect: the paper itself, dated March 2, 2021, already proves these statements and even compares its bound to the then-recent 0.96399 bound of Kleptsyn–Pollicott–Vytnova, which it improves substantially .