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On the one hand, we have the bound $|X|\\ll_{\\alpha}N^{\\alpha}$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\\ll_{\\alpha}N^{O(\\alpha^{2014})}$ for small $\\alpha$). 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Given a subset $X\\subset [N]$ and $\\alpha\\in (0,1/2)$. Suppose that $|X\\pmod p|\\leq (\\alpha+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\\ll_{\\alpha}N^{\\alpha}$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\\ll_{\\alpha}N^{O(\\alpha^{2014})}$ for small $\\alpha$). 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