POISSON APPROXIMATION AND WEIBULL ASYMPTOTICS IN THE GEOMETRY OF NUMBERS

Both the paper and the model prove Poisson convergence of exceedances and Weibull limits under the same separation hypothesis and (a,b,c)-regularity. However, the model’s evaluation of the limiting constant m0 conflicts with the paper’s. With the scaling δn=|Fn|^{-1/a}(log|Fn|)^{-b/a}, the paper derives |Fn|·Vold(C(δn u))→c·u^a·a^b, hence Nn(·;δn u)⇒ Poi((c a^b/(2ζ(d))) u^a) and η̃Fn/δn⇒Wei((2ζ(d))^{1/a}/(c a^b)^{1/a},a) under µd, for d≥3. The model instead claims m0=c(1/a)^b/(2ζ(d)), attributing the discrepancy to “normalization,” but this would contradict the paper’s explicit computation and statements. Apart from this constant, the model’s proof sketch (small-target asymptotics, mixing, factorial moments) aligns conceptually with the paper’s framework, though the paper implements a more robust W-equidistribution criterion and precise second-moment bounds via Siegel–Rogers.