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In this paper we show that if $\\alpha > 0$, there exists a set $E\\in F_\\alpha$ such that $\\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\\alpha}\\log^{-\\theta}(\\frac{1}{x})$, $\\theta>\\frac{1+3\\alpha}{2}$, which improves on the the previously known bound, that $H^{\\beta}(E) = 0$ for $\\beta>1/2+3/2\\alpha$. 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