HAUSDORFF DIMENSION OF THE PARAMETERS FOR (α, β)-TRANSFORMATIONS WITH THE SPECIFICATION PROPERTY

The paper proves that for a fixed endpoint coding u with u0=0, u≼σn u and K:=sup u_i<∞, the set of β>K+2 for which the (α(β),β)-shift has specification has Hausdorff dimension 1. Its proof constructs, for each large N, a Cantor family EN of parameters via an IFS on a coding map φ, invokes a specification criterion (bounded D(u),D(v) for β>2), and transfers dimension from φ(EN) to EN using a Lipschitz estimate (Lemma 3.2), yielding dimH EN ≥ log(N−3)/log N → 1 as N→∞, hence the target set has dimension 1. The candidate solution follows the same overall scheme (fix u; build a large class of v avoiding extreme digits; use a specification criterion; create a Cantor set of parameters with an IFS; obtain a dimension lower bound tending to 1). However, it informally places the IFS directly on the β-parameter line and claims “uniform contraction/branching” there; the paper instead defines an IFS on φ(EN) and uses a Lipschitz map β↦φ(β) to transfer dimension back to EN. Despite this mismatch in where the IFS acts, the candidate’s argument reaches the correct conclusion and is substantially the same in spirit. Minor issues in the paper (a typographical derivative formula for α(β) missing the coefficients u_j) do not affect correctness. Key steps and statements are explicit in Theorem 1, Theorem 2.2, Lemma 3.1, and the construction of EN and φ(EN).