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For instance, one can show that if $\\emptyset \\neq K \\subset \\mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \\geq 1$, then there exists a point $x_{0} \\in K$ such that $\\dim_{\\mathrm{p}} K \\cdot (K - x_{0}) = 1$. 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For instance, one can show that if $\\emptyset \\neq K \\subset \\mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \\geq 1$, then there exists a point $x_{0} \\in K$ such that $\\dim_{\\mathrm{p}} K \\cdot (K - x_{0}) = 1$. 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