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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 ?

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I know that this is not necessary to state the question, but why don't you add some definitions, perhaps references to articles you have already looked at or even a little motivation? – Gjergji Zaimi Sep 9 '10 at 6:35
I mostly agree with Gjergji, except that I think that it is not only advisable but definitely necessary to state the definitions, so that the question makes sense. – Victor Protsak Sep 9 '10 at 7:02
Are self-affine tiles a generalisation of rep-tiles? Are they the same in dimension 1? – Henry Segerman Sep 9 '10 at 7:15
In the paper of Kenyon mentioned above he showed that all prototiles which can be used to give a self-replicating tiling are necessarily self-affine tiles. On the other hand Theorem 1.2 of the Lagarias wang paper says any self-affine tile can be used as a prototile to give a self-replicating tiling of $\mathbb R^n$ – Vagabond Sep 9 '10 at 7:31
@ Gjergji Zaimi and Victor Prostak made the changes as you have recommended. Please let me know if I should add more details. – Vagabond Sep 9 '10 at 7:33
up vote 2 down vote accepted

The classification is given in section 5 of "Integral Self-Affine Tiles in $\mathbb R^n$ I. Standard and Nonstandard Digit Sets" 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".

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Is there a classification of tiling sets associated with a self affine set which gives a self-replicating tiling of $R^n$ ? Is there some way to uniquely associate such a tiling set with a self-affine set ( I am asking about self-replicating tiling ofcourse). Can you suggest some text/ review paper where I can find a good updated survey of self replicating tiling by self-affine sets. Specially the number theoretic aspects ? – Vagabond Sep 9 '10 at 8:37
I'm not sure, but if you refer to results of tilings up to $\mathbb Z$ similarity then some work has been done, see for example – Gjergji Zaimi Sep 9 '10 at 8:55
You may find useful J.P. Gabardo, X.Yu, Natural tiling, lattice tiling and Lebesgue measure of integral self-affine tiles, J. Lond. Math. Soc., II. Ser. 74 (1) (2006) 184-204. – Ievgen Bondarenko Sep 9 '10 at 9:15

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