QUANTITATIVE RECURRENCE PROPERTIES AND STRONG DYNAMICAL BOREL-CANTELLI LEMMA FOR DYNAMICAL SYSTEMS WITH EXPONENTIAL DECAY OF CORRELATIONS

The paper proves the self-return shrinking-target strong Borel–Cantelli law under Conditions I–V by (i) obtaining correlation bounds for sets with regular boundaries, (ii) introducing scaled targets Ên so that expectations are comparable to γn, (iii) verifying Sprindžuk’s correlation criterion, and (iv) using Zygmund differentiation to pass from Ên back to the original self-centered rectangles. The candidate solution misapplies the SBC lemma to a sequence whose expectations depend on x, uses an unsupported rank-one decomposition on X×X, and gives a flawed convergence estimate of ∑ µ×µ(D(rn)).