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Let $\\nu$ be a Borel probability measure on $S^{1}$ satisfying $\\nu(B(x,r)) \\leq Cr^{\\epsilon}$ for all $x \\in S^{1}$ and $r > 0$. Let $K \\subset B(1) \\subset \\mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. 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Let $\\nu$ be a Borel probability measure on $S^{1}$ satisfying $\\nu(B(x,r)) \\leq Cr^{\\epsilon}$ for all $x \\in S^{1}$ and $r > 0$. Let $K \\subset B(1) \\subset \\mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. 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