bibkey: simon2007equilibrium authors: Barry Simon year: 2007 title: “Equilibrium Measures and Capacities in Spectral Theory” doi: null url: https://arxiv.org/abs/0711.2700v1 claim: “Theorems 1.10 and 1.11 and Appendix A, Equation (A.8), give regular-measure norm asymptotics and the capacity of an interval.” strata_touched:

• D5/S3/Constants/Moments/CatalanSquareHankelLimits license: citation-only triage: anchor

Simon’s regular-measure route to the same rate

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The supplied primary-source audit read https://arxiv.org/abs/0711.2700v1 and identifies Theorem 1.10(i), the Erdos-Turan criterion in Theorem 1.11, and Appendix A, Equation (A.8). The first relates regularity to the asymptotic norms of monic orthogonal polynomials. The second makes a measure regular on an interval when its density is positive almost everywhere there. Equation (A.8) gives the ordinary logarithmic capacity of an interval; for [0,16] it is 4.

Applied to the positive product density from Lin’s route, multiplication by z^r for r=1,2 preserves positivity almost everywhere on (0,16) and therefore preserves regularity. The standard monic Gram determinant identity then expresses each shifted Hankel determinant as the product of the squared monic orthogonal-polynomial norms. Summing their logarithmic asymptotics gives log D_r(n)/n^2 -> log 4 = 2 log 2 for both shifts.

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This is an attributed ordinary classical corollary of the cited results, not a claim that Simon states A277829 or A278770. The repository proof is a separate elementary argument: it constructs explicit Gram models and matching Chebyshev upper and lower estimates without taking regular-measure theory as a Lean premise. The theorem locators are inherited from the supplied source audit; this metadata pass did not independently reread the paper.