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We prove that there exists a constant $\\delta>0$ such that, for every injective sequence $(s_i)_{i\\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\\subseteq X$ whose distinct points $x,x'$ satisfy \\[ \\liminf_{i\\to\\infty}\\rho(s_i x,s_i x')=0, \\qquad \\limsup_{i\\to\\infty}\\rho(s_i x,s_i x')>\\delta. \\] The method also yields higher-order scrambled Cantor sets. 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We prove that there exists a constant $\\delta>0$ such that, for every injective sequence $(s_i)_{i\\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\\subseteq X$ whose distinct points $x,x'$ satisfy \\[ \\liminf_{i\\to\\infty}\\rho(s_i x,s_i x')=0, \\qquad \\limsup_{i\\to\\infty}\\rho(s_i x,s_i x')>\\delta. \\] The method also yields higher-order scrambled Cantor sets. 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