A SHRINKING TARGET THEOREM FOR IETS

The paper proves Theorem 2.5 via a geometric second-moment/Borel–Cantelli scheme and then upgrades a positive-measure limsup set to full measure using ergodicity of the product R_α × f (for a.e. α) and the no-common-eigenvalues criterion, yielding λ(limsup_i f^{-i}(B(R_α^i x, b_i))) = 1 for a.e. x, α . The candidate solution gives a simpler, independent proof: for each fixed y, the events {f^i(y) ∈ B(R_α^i x, b_i)} are pairwise independent in (x, α); since ∑_i λ(B(0, b_i)) diverges, Kochen–Stone implies occurrence infinitely often with probability 1, and Tonelli/Fubini then yields the theorem for a.e. x, α. Aside from a cosmetic “factor 2” slip in the ball-length bound, the model’s argument is correct and in fact removes the paper’s use of ergodicity and monotonicity. Hence both arguments are correct, with the model presenting a shorter, stronger proof while the paper’s proof is also valid as written.