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In this paper, we prove that for every integer $\\ell\\ge2$, there are $\\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $\\Gamma$ is $(1-\\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\\ell}n^{1+1/\\ell}$ edges, then $\\Gamma$ contains at least $C'n^2t^{2\\ell-2}$ induced copies of $C_{2\\ell}$. 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In this paper, we prove that for every integer $\\ell\\ge2$, there are $\\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $\\Gamma$ is $(1-\\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\\ell}n^{1+1/\\ell}$ edges, then $\\Gamma$ contains at least $C'n^2t^{2\\ell-2}$ induced copies of $C_{2\\ell}$. 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