Fekete convergence from balanced integer predecessors in shrinking corridors

Abstract

Let w be positive, measurable and nonincreasing, with divergent square integral. Consider real sequences satisfying exact subadditivity only for integer pairs n≤ m≤ n+n w(log n). We replace eventual legality of adjacent pairs by a weaker condition: every sufficiently large output has a legal decomposition whose two summands differ by O(N^α) for some α<1, or by O(N/(log N)^(3/2+ε)) for some ε>0. Under either condition, a_n/n converges in ℝ∪{-∞}, without sign, monotonicity or growth assumptions. The proof regularizes the sequence by growing local maxima, proves exact integer corridor inequalities for that majorant, and separately transfers its limit back by propagation of low values along dyadic orbits. An explicit staircase width has arbitrarily long windows of illegal adjacent pairs, satisfies the logarithmic condition, and fails every fixed power condition. Both this convergence example and an oscillating isolated-output example can have divergent integrals of every finite positive power of w. Thus analytic width budgets and integer connectivity play different roles.