S-INTEGRAL PREPERIODIC POINTS FOR MONOMIAL SEMIGROUPS OVER NUMBER FIELDS

The paper proves the uniform, effective finiteness result for monomial semigroups (Theorem 1.5) via: (i) a structural description of preperiodic points (binomial factorization) with degree lower bounds, (ii) explicit lower bounds for linear forms in logarithms (archimedean and non-archimedean) to control local distances (Theorem 5.1), and (iii) quantitative equidistribution for sequences drawn from the semigroup to compare adelic averages with a canonical height, culminating in a product-formula contradiction and a bound on degrees (Section 6). The candidate solution uses the same essential strategy—preperiodicity gives a fixed point after a prefix, linear/logarithmic bounds, quantitative equidistribution, and a product-formula summation—arriving at the same conclusion. Differences are largely presentational or technical (e.g., introducing an auxiliary set S0 of good reduction and a conjugation to a pure power map, and a brief ‘bookkeeping’ step to pass from fixed points back to α). The paper’s details and references are tighter (e.g., the binomial factorization Proposition 4.1 and the quantitative equidistribution framework), but no fatal conflict exists. See Theorem 1.5 for the statement, Proposition 4.1 and Lemma 4.2 for the structure and local size bounds, Theorem 5.1 for local distance control, and the Section 6 argument for the global contradiction and degree bound .