Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-Similar One-Dimensional Quasilattices   We study 1D quasilattices, especially self-similar ones that can be used to
generate two-, three- and higher-dimensional quasicrystalline tesselations that
have matching rules and invertible self-similar substitution rules (also known
as inflation rules) analogous to the rules for generating Penrose tilings. The
lattice positions can be expressed in a closed-form expression we call
\emph{floor form}: $x_{n} = S (n-\alpha) + (L-S) \lfloor \kappa (n-\beta)
\rfloor$, where $L > S > 0$ and $0 < \kappa < 1$ is an irrational number. We
describe three equivalent geometric constructions of these quasilattices and
show how they can be subdivided into various types of equivalence classes: (i)
\emph{lattice equivalent}, where any two quasilattices in the same lattice
equivalence class may be derived from one another by a local decoration/gluing
rule; (ii) \emph{self-similar}, a proper subset of lattice equivalent where, in
addition, the two quasilattices are locally isomorphic; and (iii)
\emph{self-same}, a proper subset of self-similar where, in addition, the two
quasilattices are globally isomorphic (i.e. identical up to rescaling). For all
three types of equivalence class, we obtain the explicit transformation law
between the floor form expression for two quasilattices in the same class. We
tabulate (in Table 1 and Figure 5) the ten special self-similar 1D
quasilattices relevant for constructing Ammann patterns and Penrose-like
tilings in two dimensions and higher, and we explicitly construct and catalog
the corresponding self-same quasilattices. Finally, we sketch the extension of
our results from degree two to degree $N$ (i.e to 1D quasilattices built from
$N$ different intervals).

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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga