Sieve dimension of union of two sets.

Question

Let $P$ be a set of primes $\leq p$. Let $A$ be a set of all integers $\leq x$ in which the elements in $A$ would avoid two classes mod $p_i$ for all $p_i \leq p$ (except $2$,$3$). My understanding from my brief reading is that this corresponds to sifting dimension two. Suppose $B$ is another set with intergers $\leq x$ which avoids two classes mod $p_i$ for all $p_i\leq p$ (except $2$,$3$), and $A \cap B$ avoids $3$ classes mod $p_i\leq p$ (except $2$,$3$). Can we estimate the set $A \cup B$?

Answer 1

The counting function of $A$ (the number of elements of $A$ less than $x$) will typically be a constant times $x/\log^2x$ (and sieves will provide an upper bound of this magnitude, though not always a lower bound). Similarly for $B$. So $A\cup B$ will have counting function that is some other constant times $x/\log^2x$. In fact it will be the sum of the other two counting functions (asymptotically), since the counting function of the intersection has magnitude $x/\log^3x$.

Maybe I'm not understanding your question well; but it seems like there's no natural way to study $A\cup B$ in this context other than studying $A$ and $B$ separately.