Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

In the paper, we focus on the connectedness of planar self-affine sets T(A,D) generated by an integer expanding matrix A with |det(A)|=3 and a collinear digit set D={0,1,b}v, where b>1 and v∈R2 such that {v,Av} is linearly independent. We discuss the domain of the digit b to determine the connectedness of T(A,D). Especially, a complete characterization is obtained when we restrict b to be an integer. Some results on the general case of |det(A)|>3 are obtained as well.