Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integer Complexity: Representing Numbers of Bounded Defect   Define $\|n\|$ to be the complexity of $n$, the smallest number of ones
needed to write $n$ using an arbitrary combination of addition and
multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$.
Based on this, this author and Zelinsky defined the "defect" of $n$,
$\delta(n):=\|n\|-3\log_3 n$, and this author showed that the set of all
defects is a well-ordered subset of the real numbers. This was accomplished by
showing that for a fixed real number $r$, there is a finite set $S$ of
polynomials called "low-defect polynomials" such that for any $n$ with
$\delta(n)<r$, $n$ has the form $f(3^{k_1},\ldots,3^{k_r})3^{k_{r+1}}$ for some
$f\in S$. However, using the polynomials produced by this method, many
extraneous $n$ with $\delta(n)\ge r$ would also be represented. In this paper
we show how to remedy this and modify $S$ so as to represent precisely the $n$
with $\delta(n)<r$ and remove anything extraneous. Since the same polynomial
can represent both $n$ with $\delta(n)<r$ and $n$ with $\delta(n)\ge r$, this
is not a matter of simply excising the appropriate polynomials, but requires
"truncating" the polynomials to form new ones.

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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph