A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such that the affinely inflated copy $A(T)$ of $T$ can be perfectly tiled with essentially disjoint translates of $T$. 


Thus we have $$ A(T) = \cup\_{i=1}^m (T+d\_i); \mathcal D= d\_1,d\_2,\dots,d\_m $$


where $|det(A)| =|\mathcal D|= m$ 


Results of Kenyon (Projecting the one-dimensional Sierpinski gasket Projecting the one-dimensional Sierpinski gasket. 
Israel J. Math. 97 (1997), 221--238.) and Lagarias and Wang (Self-affine tiles in $R^n$. Adv. Math. 121 (1996), no. 1, 21--49) tells that such sets always can be used to give a translational tiling of $R^n$ and has boundary of measure zero and has nonempty interiors.


Thus in one dimension we can think of them as a union of intervals (possibly infinitely many ).


My question is :- 


Is there a characterization of self-affine tiles in $\mathbb R$ which are union of finitely many intervals ? 



A question about  self-affine tiles 
The classification is given in section 5 of ["Integral Self-Affine Tiles in $\mathbb R^n$ I. Standard and Nonstandard Digit Sets"](http://jlms.oxfordjournals.org/content/54/1/161.abstract) by Lagarias and Wang (Theorem 5.2 and corollary 5.2a). Their result builds on the previous paper by A. M. Odlyzko, "Non-negative digit sets in positional number systems".


