A MEASURE ESTIMATE IN GEOMETRY OF NUMBERS AND IMPROVEMENTS TO DIRICHLET’S THEOREM

The paper proves a weighted zero–one law (Theorem 1.2) with the exact summation criterion Σ k^{-1} Fψ(k)^{κd} log^{λd}(1/Fψ(k)) and an additional quasi-independence condition (1.11) in the divergence case, and supplies the key asymptotic measure estimate µd(Δ̃r) ≍ r^{κd} log^{λd}(1/r) (Theorem 5.1) needed for the Borel–Cantelli analysis; see Theorem 1.2 and surrounding discussion, and Theorem 5.1 on thickened targets . The model’s blueprint mirrors the dynamical strategy but claims the necessary volume asymptotics were not available and labels the problem “likely open”; this is contradicted by the paper’s explicit construction and asymptotics, together with effective equidistribution and doubly mixing to handle dependence (equations (1.14), (1.15)) .