A NEW CLASS OF α-FAREY MAPS AND AN APPLICATION TO NORMAL NUMBERS

The uploaded paper proves that the set of α-normal numbers equals the set of 1-normal numbers (Theorem 4), using an isomorphism between the natural extensions of α-Farey maps and the classical Farey map, together with a careful induced-system counting argument that explicitly controls unbounded return times via the N̂/N normalization; see the statement of Theorem 4 and the definitions of α-normality, as well as the key counting step (31) in Section 5.1 . By contrast, the candidate solution posits a general ‘synchronizing finite-state recodings preserve normality’ lemma under a bounded-distortion hypothesis (uniformly bounded output length per input symbol), and then asserts that α-Farey inducing yields such a recoding between α- and 1-digit sequences. This bounded-length assumption is not supported for 0<α<1/2: inducing produces unbounded return times in general, so the required uniform bound fails. The paper’s proof explicitly handles the lack of uniform boundedness via induced-map normalization rather than a uniformly bounded code . Hence, while the candidate reaches the correct conclusion, its proof relies on an unjustified (and generally false) uniform bounded-length assumption for the inducing-based recoding; the paper’s argument is correct.