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We prove that, for $N$ sufficiently large, $$ \\frac{1}{10}\\frac{d}{N^{\\frac{1}{2} + \\frac{1}{2d}}} \\leq \\mathbb{E} D_N^*(\\mathcal{P}) \\leq \\frac{\\sqrt{d} (\\log{N})^{\\frac{1}{2}}}{N^{\\frac{1}{2} + \\frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. 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We prove that, for $N$ sufficiently large, $$ \\frac{1}{10}\\frac{d}{N^{\\frac{1}{2} + \\frac{1}{2d}}} \\leq \\mathbb{E} D_N^*(\\mathcal{P}) \\leq \\frac{\\sqrt{d} (\\log{N})^{\\frac{1}{2}}}{N^{\\frac{1}{2} + \\frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. 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