New dimension bounds for αβ sets

The paper proves Theorem 1.1 rigorously: for τ1, τ2 ≥ 2 with 2τ1 < τ2 + 2, α ∈ E(τ1) and β ∈ W(τ2), any αβ orbit (x_n) satisfies dim_B({x_n}) ≥ 1 − 2(τ1 − 1)/τ2. The argument uses a dichotomy: either the α/β step counts are not comparable and the orbit is dense (hence box dimension 1), or they are comparable; in the latter case, a continued-fraction selection and a pigeonhole argument produce Θ(q′) points that are ≥ c/q′ separated, yielding the stated lower bound on the upper box dimension (Propositions 3.1–3.2 and Lemma 2.1, culminating in Theorem 1.1) . By contrast, the candidate solution’s pair-count/energy method produces only a constant-in-r lower bound N(F_N,r) ≳ 1/( (rq)N^{(τ1−1)+ε} + 1 ), which cannot imply a positive box-dimension exponent (since log of a constant divided by −log r → 0). It also imposes an incorrect threshold r ≥ N q^{1−τ2} instead of r ≥ N|ε| = N q^{−τ2}, and misidentifies the “balanced” choice θ = τ2/2 − 1 as the condition rq ≍ 1. Hence the model’s proof is flawed and does not establish the claimed dimension lower bound, while the paper’s proof is correct.