For m=1,2,3, finite integral of (1-cos theta)^m log w implies (S-1)^m alpha in l^2 and alpha in l^{2m+2}.
Du, Sum rules and Simon spectral gem problem on higher order Szegő theorems, arXiv preprint(2023)
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Single-Point Higher-Order Szeg\H{o} Sum Rules in OPUC: Necessity for $m=1,2,3$
For m=1,2,3, finite integral of (1-cos theta)^m log w implies (S-1)^m alpha in l^2 and alpha in l^{2m+2}.