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Precisely, for an increasing function $\\varphi: \\mathbb{N} \\rightarrow \\mathbb{N}$, one is interested in the Hausdorff dimension of the sets\\[E\\_\\varphi = \\left\\{x\\in (0,1): \\lim\\_{n\\to\\infty} \\frac {S\\_n(x)} {\\varphi(n)} =1\\right\\}.\\]Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. 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