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For any $n \\geq 1$, write $T_n(x) = \\max_{1 \\leq k \\leq n}\\{a_k(x)\\}$. We are interested in the Hausdorff dimension of the fractal set \\[ E_\\phi = \\left\\{x \\in (0,1): \\lim_{n \\to \\infty} \\frac{T_n(x)}{\\phi(n)} =1\\right\\}, \\] where $\\phi$ is a positive function defined on $\\mathbb{N}$ with $\\phi(n) \\to \\infty$ as $n \\to \\infty$. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. 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