Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost all orbits of the Collatz map attain almost bounded values   Define the \emph{Collatz map} $\mathrm{Col} : \mathbb{N}+1 \to \mathbb{N}+1$
on the positive integers $\mathbb{N}+1 = \{1,2,3,\dots\}$ by setting
$\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even,
and let $\mathrm{Col}_{\min}(N) := \inf_{n \in \mathbb{N}} \mathrm{Col}^n(N)$
denote the minimal element of the Collatz orbit $N, \mathrm{Col}(N),
\mathrm{Col}^2(N), \dots$. The infamous \emph{Collatz conjecture} asserts that
$\mathrm{Col}_{\min}(N)=1$ for all $N \in \mathbb{N}+1$. Previously, it was
shown by Korec that for any $\theta > \frac{\log 3}{\log 4} \approx 0.7924$,
one has $\mathrm{Col}_{\min}(N) \leq N^\theta$ for almost all $N \in
\mathbb{N}+1$ (in the sense of natural density). In this paper we show that for
\emph{any} function $f : \mathbb{N}+1 \to \mathbb{R}$ with $\lim_{N \to \infty}
f(N)=+\infty$, one has $\mathrm{Col}_{\min}(N) \leq f(N)$ for almost all $N \in
\mathbb{N}+1$ (in the sense of logarithmic density). Our proof proceeds by
establishing an approximate transport property for a certain first passage
random variable associated with the Collatz iteration (or more precisely, the
closely related Syracuse iteration), which in turn follows from estimation of
the characteristic function of a certain skew random walk on a $3$-adic cyclic
group at high frequencies. This estimation is achieved by studying how a
certain two-dimensional renewal process interacts with a union of triangles
associated to a given frequency.

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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat