Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classification of tile digit sets as product-forms   Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A
fundamental question in the fractal tiling theory is to understand the
structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral
self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In
our previous paper, we classified such tile digit sets ${\mathcal
D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into
product of cyclotomic polynomials. In this paper, we first show that a tile
digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal
D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This
allows us to combine the technique of Coven and Meyerowitz on integer tiling on
${\Bbb R}^1$ together with our previous results to characterize explicitly all
tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$
distinct primes) as {\it modulo product-form} of some order, an advance of the
previously known results for $A = p^\alpha$ and $pq$.

24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex