ON GOOD APPROXIMATIONS AND BOWEN-SERIES EXPANSION

The paper’s Theorem 4.1 states and proves (i) the two-sided estimate 1/(|Wr|+2µ) ≤ D(ζr)^2 |α−ζr| ≤ 1/|Wr| and (ii) a Legendre-type criterion asserting the existence of ε0>0 with D(ζ)^2|α−ζ|<ε0 forcing ζ to be a convergent with |Wr|>0; see the statement and proof outline in Theorem 4.1 and §4.2.1–4.2.2, together with the denominator and horoball facts (1.6) and (A.3) . The candidate solution proves the same two items by a slightly different route: it derives the exact identity D(ζr)^2|α−ζr|=1/|xr+ρr| and bounds |xr+ρr| against |Wr| and |Wr|+2µ via cusp coordinates, then proposes an explicit ε0=1/(2µ) for the Legendre-type part. The first part aligns closely with the paper’s 2T-based argument (since 2T=|xr+ρr|), while the second part supplies a concrete (though not claimed optimal) threshold; the paper only requires existence of ε0, established via separation and reduced-form arguments (Steps (0)–(3)) .