Recent progress on pointwise normality of self-similar measures

The paper proves Theorem 1.2: any non-atomic self-similar Rajchman measure is pointwise absolutely normal, via a conditional-measure/partition method (Theorem 2.1) and a Fourier-mode criterion (Lemma 3.1), then shows the Rajchman property forces those Fourier modes to vanish uniformly after suitable scaling; see Theorem 1.2, Lemma 3.1, and the conclusion of Section 3 (including the use of the identity \hat T_b^n \circ c = c \circ T_b^n for integer b) . The candidate solution instead attempts a direct van der Corput/L^2 approach and then (incorrectly) applies Chebyshev plus Borel–Cantelli to a tail sup over N. That step is invalid: one cannot bound μ({sup_{N≥N_j}|S_N|>a}) by sup_{N≥N_j} E|S_N|^2/a^2, and a union bound would require summability over N, which fails for the obtained O(1/N) L^2 bounds. Moreover, the candidate’s method, if correct, would imply the (false) general statement that every Rajchman measure is pointwise absolutely normal, contradicting Lyons’ negative answer to the Kahane–Salem question cited in the paper .