An interesting question on infinite arithmetic progressions

I was working on a problem that consisted of deciding if the language a finite automaton (the alphabet of which is ${0,1}$ and the words accepted are binary encoded positive integers) contains an infinite arithmetic progression, and I bumped into the following problem:

Say we have a finite bases set of natural numbers $B= {n_1, dots, n_k}$ and a finite set of tuples $T={(m_1, a_1), dots, (m_t, a_t)}$, can we know if the following set $S$ contains an infinite arithmetic progression:
$$S = { s mid s = n_i*m_j + a_j text{ for some } 1geq i geq k text{ and } 1geq j geq t }$$.

Any ideas are appreciated 🙂