On sums and intersects of sequences

Let a,b and c be three natural numbers with ab+c and 2d∉B. In this note we prove that there exists a sequence A={an}n=1∞ with a1=d and an+1−an∈{a,b} such that (A+A)∩B=∅. On the other hand, we show that there exists a sequence {bn}n=1∞ of natural numbers with bn+1⩾2bn such that for any natural number a, (A+A)∩B≠∅ for any sequence A={an}n=1∞ of natural numbers with an+1−an∈{a,2a}.