I have a problem in mind and I want to know which research field it belongs to, then I can read something more specific(maybe).

The problem is :

consider two sequences of integers $s_n,r_n$

then consider the set of integers $A=\lbrace m|m=s_nk+r_n,k\in B\rbrace$, where $B$ is a subset of $\mathbb{Z}$

so, which condition(s) should $s_n,r_n,B$ satisfy such that $\mathbb{Z}-A$(integers not in $A$) is a finite/infinte set.

for example:

if $s_n=n,r_n=0,B=\mathbb{Z}-\lbrace 0,1,-1 \rbrace$ then $\mathbb{Z}-A$ which is the set of integer primes is infinite.

More general, we can consider more pairs of sequences, $(s_n^1,r_n^1)...(s_n^L,r_n^L)$ and consider $A^L=\lbrace m|m=s_n^Lk+r_n^L,k\in B^L\rbrace$

then when can the set $\mathbb{Z}-A^1-A^2-...-A^L$ be finite/infinite

of course we can also ask for the distributions of $\mathbb{Z}-A$ like the prime number theorem.

So again, I'm not asking for the solutions of this problem, because indeed I think it is a very large topic in number theory, but I do not know some terminologies, anyone knows anything about this problem, like which research fields of number theory it belongs to, does it have any special name?

Thx :D

• $s_n=1$, $k=2$ is in $B$, so $2$ is in $A$, as is every prime number (and every composite number, of course). Anyway, I think you might be interested in the concept of "covering congruences". – Gerry Myerson Mar 12 '15 at 22:54
• @GerryMyerson thanks, that's helpful :DD – Kuhn Mar 13 '15 at 12:59
• I think it would be easier to point out connections to the literature if more specifics of $B$ and sequences $s_n,r_n$ were given. – hardmath Mar 16 '15 at 14:35