RATIONAL NUMBERS IN ×b-INVARIANT SETS

The uploaded paper proves (i) finiteness of S-integers in any non-dense A with Tb(A ∩ Q) ⊆ A (Theorem 1.2) and (ii) quantitative lower bounds for P(d) of rationals a/d ∈ A (Theorem 1.6) via a clean orbit-structure lemma (Theorem 2.4) and elementary estimates on ord(b, d), culminating in log(2εd) ≪ P(d)^2 log b / log P(d) and the stated cases P(d) ≳ sqrt((1/log b) log(2εd)) and P(d) ≳ sqrt((1/log b) log(2εd) log log(2εd)) when P(d) < b and > b respectively . By contrast, the model’s outline relies on a heavy Fourier/exponential-sum approach and makes key mistakes: (a) it sums one b-adic cylinder per level and incorrectly treats ∑|U_k| ≍ K, whereas the main term scales like ∑ (|U_k|/d) ≍ ∑ b^{-k} = O(1), so no K-gain is available; (b) it asserts a nonstandard Burgess-driven improvement S(q) ≪ √q/√log K uniformly in the subgroup-additive sums without a valid reduction; and (c) it attempts to deduce a lower bound on P(d) from an upper bound on ord_q(b) without any lower control on ord_q(b). The paper’s argument is coherent and complete; the model’s is not.