Hausdorff and Box Dimension Bounds for Subfractals Induced by S-Gap Shifts

The paper proves that for subfractals induced by S‑gap shifts, the Hausdorff and both box dimensions lie between the unique zeros h and H of the lower/upper pressure functions P and P̄, with a complete argument tailored to S‑gap language via a core/prefix/suffix decomposition and an auxiliary Q(t), plus a careful covering/counting argument that yields N_δ(F_{X(S)}) ≤ C δ^{-H} (Theorem 4.1; see the setup, Definition 3.1, and the proof outline using U_r/U_δ and bounds (4)–(5) in the text) . By contrast, the candidate solution’s lower bound and Hausdorff upper bound are standard and broadly correct, but its upper box-dimension step misuses a stopping-time cover: it asserts |W̄(r)| ≤ r^{-t}∑(D c̄_ω)^t, which is the wrong inequality direction, and it omits the needed lower bound c̄_ω ≳ r to deduce |W̄(r)| ≲ r^{-t+α}. The paper’s argument correctly supplies the needed counting via uniform bounds on ∑_{ω∈L_n} c̄_ω^H and derives the box-dimension bound directly . The model also assumes, without justification for arbitrary subshifts, the existence/regularity of the pressure limits and uniqueness of zeros; the paper addresses this near the zero by relating P to Q(t) for S‑gap shifts .