The following version of a theorem of Ekedahl, known as the Ekedahl sieve, can be found in this paper by Bhargava and Shankar for example, is stated as follows: let $B$ be a compact subset of $mathbb{R}^n$ having finite measure, and let $Y$ be any closed subscheme of $mathbb{A}^n(mathbb{Z})$ having co-dimension $k geq 1$. Let $r, M$ be positive real numbers. Then

$$displaystyle # {mathbf{a} in rB cap mathbb{Z}^n : mathbf{a} pmod{p} in Y(mathbb{F}_p) text{ for some prime } p > M } $$

$$ = O left(

frac{r^n}{M^{k-1} log M} + r^{n-k+1} right)$$

with the implied constant depending only on $Y$ and $M$.

If we replace the set on the left hand side with the set

$$displaystyle S_M(r) = {mathbf{a} in rB cap mathbb{Z}^n : mathbf{a} pmod{p} in Y(mathbb{F}_p) text{ for all } p | q text{ for some square-free } q > M }$$

can we obtain an upper bound of comparable strength to the original theorem?