Abstract. This paper is motivated by the following question in sieve theory. Given a subset X ⊂ [N ] and α ∈ (0, 1/2). Suppose that |X (mod p)| ≤ (α + o(1))p for every prime p. How large can X be? On the one hand, we have the bound |X| ≪ α N α 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| ≪ α N O(α 2014 ) for small α). The result follows from studying the average size of |X (mod p)| as p varies, whenis the value set of a polynomial f (x) ∈ Z[x].