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We introduce an arithmetically defined index $\\delta (\\alpha, \\theta)$ and show that for $\\alpha\\notin\\mathbb{Q},\\,$ $\\sigma_{sc}(h_{\\lambda,\\alpha,\\theta})=\\overline{\\{e:\\gamma_{\\lambda}(e) <\\delta (\\alpha, \\theta) \\}}$ and $\\sigma_{pp}(h_{\\lambda,\\alpha,\\theta})=\\{e:\\gamma_{\\lambda}(e) \\geq \\delta (\\alpha, \\theta) \\}$. 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