Expanding Polynomials and Connectedness of Self-Affine Tiles

Little is known about the connectedness of self-affine tiles in ${\Bbb R}^n$. In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the {\it height reducing property}, on expanding polynomials (i.e., all the roots have moduli $ > 1$), we show that all such tiles in ${\Bbb R}^n, n \leq 3$, are connected. The problem is still unsolved for higher dimensions. For this we make another investigation on this algebraic criterion. We improve a result of Garsia concerning the heights of expanding polynomials. The new result has its own interest from an algebraic point of view and also gives further insight to the connectedness problem.