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Almost all orbits of the Collatz map attain almost bounded values

Almost all orbits of the Collatz map attain almost bounded values <jats:title>Abstract</jats:title> <jats:p>Define the <jats:italic>Collatz map</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline1.png" /> <jats:tex-math> ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on the positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline2.png" /> <jats:tex-math> $\mathbb {N}+1 = \{1,2,3,\dots \}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by setting <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline3.png" /> <jats:tex-math> ${\operatorname {Col}}(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> equal to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline4.png" /> <jats:tex-math> $3N+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:italic>N</jats:italic> is odd and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline5.png" /> <jats:tex-math> $N/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:italic>N</jats:italic> is even, and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline6.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the minimal element of the Collatz orbit <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline7.png" /> <jats:tex-math> $N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The infamous <jats:italic>Collatz conjecture</jats:italic> asserts that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline8.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N)=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline9.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Previously, it was shown by Korec that for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline10.png" /> <jats:tex-math> $\theta&gt; \frac {\log 3}{\log 4} \approx 0.7924$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, one has <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline11.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) \leq N^\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for almost all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline12.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (in the sense of natural density). In this paper, we show that for <jats:italic>any</jats:italic> function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline13.png" /> <jats:tex-math> $f \colon \mathbb {N}+1 \to \mathbb {R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline14.png" /> <jats:tex-math> $\lim _{N \to \infty } f(N)=+\infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, one has <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline15.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) \leq f(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for almost all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline16.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline17.png" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic cyclic group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline18.png" /> <jats:tex-math> $\mathbb {Z}/3^n\mathbb {Z}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> 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.</jats:p> http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum of Mathematics, Pi CrossRef

Almost all orbits of the Collatz map attain almost bounded values

Forum of Mathematics, Pi , Volume 10 – Jan 1, 2022

Almost all orbits of the Collatz map attain almost bounded values


Abstract

<jats:title>Abstract</jats:title>
<jats:p>Define the <jats:italic>Collatz map</jats:italic><jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline1.png" />
<jats:tex-math>
${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> on the positive integers <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline2.png" />
<jats:tex-math>
$\mathbb {N}+1 = \{1,2,3,\dots \}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> by setting <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline3.png" />
<jats:tex-math>
${\operatorname {Col}}(N)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> equal to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline4.png" />
<jats:tex-math>
$3N+1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> when <jats:italic>N</jats:italic> is odd and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline5.png" />
<jats:tex-math>
$N/2$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> when <jats:italic>N</jats:italic> is even, and let <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline6.png" />
<jats:tex-math>
${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> denote the minimal element of the Collatz orbit <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline7.png" />
<jats:tex-math>
$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. The infamous <jats:italic>Collatz conjecture</jats:italic> asserts that <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline8.png" />
<jats:tex-math>
${\operatorname {Col}}_{\min }(N)=1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for all <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline9.png" />
<jats:tex-math>
$N \in \mathbb {N}+1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Previously, it was shown by Korec that for any <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline10.png" />
<jats:tex-math>
$\theta&gt; \frac {\log 3}{\log 4} \approx 0.7924$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, one has <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline11.png" />
<jats:tex-math>
${\operatorname {Col}}_{\min }(N) \leq N^\theta $
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for almost all <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline12.png" />
<jats:tex-math>
$N \in \mathbb {N}+1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> (in the sense of natural density). In this paper, we show that for <jats:italic>any</jats:italic> function <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline13.png" />
<jats:tex-math>
$f \colon \mathbb {N}+1 \to \mathbb {R}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline14.png" />
<jats:tex-math>
$\lim _{N \to \infty } f(N)=+\infty $
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, one has <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline15.png" />
<jats:tex-math>
${\operatorname {Col}}_{\min }(N) \leq f(N)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for almost all <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline16.png" />
<jats:tex-math>
$N \in \mathbb {N}+1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation 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 <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline17.png" />
<jats:tex-math>
$3$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-adic cyclic group <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline18.png" />
<jats:tex-math>
$\mathbb {Z}/3^n\mathbb {Z}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> 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.</jats:p>

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Publisher
CrossRef
Copyright
© The Author(s), 2022. Published by Cambridge University Press
ISSN
2050-5086
DOI
10.1017/fmp.2022.8
Publisher site
See Article on Publisher Site

Abstract

<jats:title>Abstract</jats:title> <jats:p>Define the <jats:italic>Collatz map</jats:italic><jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline1.png" /> <jats:tex-math> ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on the positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline2.png" /> <jats:tex-math> $\mathbb {N}+1 = \{1,2,3,\dots \}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by setting <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline3.png" /> <jats:tex-math> ${\operatorname {Col}}(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> equal to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline4.png" /> <jats:tex-math> $3N+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:italic>N</jats:italic> is odd and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline5.png" /> <jats:tex-math> $N/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:italic>N</jats:italic> is even, and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline6.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the minimal element of the Collatz orbit <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline7.png" /> <jats:tex-math> $N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The infamous <jats:italic>Collatz conjecture</jats:italic> asserts that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline8.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N)=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline9.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Previously, it was shown by Korec that for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline10.png" /> <jats:tex-math> $\theta&gt; \frac {\log 3}{\log 4} \approx 0.7924$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, one has <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline11.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) \leq N^\theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for almost all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline12.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (in the sense of natural density). In this paper, we show that for <jats:italic>any</jats:italic> function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline13.png" /> <jats:tex-math> $f \colon \mathbb {N}+1 \to \mathbb {R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline14.png" /> <jats:tex-math> $\lim _{N \to \infty } f(N)=+\infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, one has <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline15.png" /> <jats:tex-math> ${\operatorname {Col}}_{\min }(N) \leq f(N)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for almost all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline16.png" /> <jats:tex-math> $N \in \mathbb {N}+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline17.png" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-adic cyclic group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508622000087_inline18.png" /> <jats:tex-math> $\mathbb {Z}/3^n\mathbb {Z}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> 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.</jats:p>

Journal

Forum of Mathematics, PiCrossRef

Published: Jan 1, 2022

There are no references for this article.