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\title{Numerical observations on extremal Fourier coefficients\\of the Syracuse random variable}
\author{Taku Toguchi\thanks{Independent researcher, Tokyo. \texttt{toguchitaku@gmail.com}}}
\date{May 18, 2026}

\begin{document}
\maketitle

\begin{abstract}
For the Syracuse random variable $\Syrac(\mathbb{Z}/3^n\mathbb{Z})$ introduced by Tao \cite{Tao2022}, we report numerical computations of its characteristic function for $1 \le n \le 21$, covering supports of size up to $3^{21} \approx 1.05 \times 10^{10}$. Writing $\psi_n(k) := \mathbb{E}\,e^{-2\pi i\, 2^k \,\Syrac/3^n}$ and $k^*(n) := \arg\max_k |\psi_n(k)|$ in a canonical fundamental domain, we observe that $k^*(n)$ exhibits an approximately linear drift with slope near $4/3$. The simple closed-form predictor $\lfloor 4(n-1)/3 \rfloor$ matches $k^*(n)$ for $n \in \{4, 5, \ldots, 17\}$ and for $n = 19$, but fails at $n \in \{18, 20, 21\}$, with $k^*(n) - \lfloor 4(n-1)/3 \rfloor$ equal to $+1$, $+1$, and $+2$ respectively. The ratio $k^*(21)/21$ equals $4/3$ exactly. The interval slope $(k^*(21) - k^*(4))/17 = 24/17 \approx 1.412$ slightly exceeds $4/3$ because of the endpoint deviation at $n = 21$. We pose the asymptotic behaviour of $k^*(n)$ as a numerical open question.
\end{abstract}

\section{Introduction}

The Collatz conjecture asks whether iterating the map $\Col(N) = N/2$ if $N$ is even and $3N+1$ if $N$ is odd, starting from any positive integer $N$, eventually reaches $1$. In a major breakthrough, Tao \cite{Tao2022} established that ``almost all'' Collatz orbits (in the logarithmic density sense) attain almost bounded values. The technical core of \cite{Tao2022} is a fine-scale equidistribution analysis of an associated probabilistic object, the \emph{Syracuse random variable}, defined as follows.

\begin{definition}[{\cite[Definition 1.7, (1.21)]{Tao2022}}]\label{def:syrac}
For each $n \ge 1$, the Syracuse random variable $\Syrac(\mathbb{Z}/3^n\mathbb{Z})$ on the cyclic group $\mathbb{Z}/3^n\mathbb{Z}$ is defined by
\begin{equation}\label{eq:syrac-def}
\Syrac(\mathbb{Z}/3^n\mathbb{Z}) \equiv \sum_{m=1}^n 3^{m-1} \cdot 2^{-a_{[1,m]}} \pmod{3^n},
\end{equation}
where $a_{[1,m]} := a_1 + \cdots + a_m$ and the $a_j$ are i.i.d.\ copies of $\Geom(2)$, i.e., $\mathbb{P}(a_j = k) = 2^{-k}$ for $k \ge 1$.
\end{definition}

A central technical estimate in \cite{Tao2022} is the following super-polynomial decay of the characteristic function:

\begin{proposition}[{\cite[Proposition 1.17]{Tao2022}}]\label{prop:tao}
Let $n \ge 1$ and $\xi \in \mathbb{Z}/3^n\mathbb{Z}$ not divisible by $3$. Then for any fixed $A > 0$,
\begin{equation}\label{eq:tao-bound}
\left| \mathbb{E}\, e^{-2\pi i \xi \Syrac(\mathbb{Z}/3^n\mathbb{Z}) / 3^n} \right| \ll_A n^{-A}.
\end{equation}
\end{proposition}

This bound is established uniformly over $\xi$ but does not, on its face, identify which $\xi$ come closest to realising the bottleneck. The present paper records numerical observations on this question, drawn from full-domain numerical computation of $\psi_n$ for $n \le 21$.

\subsection{Notation}

We write $\widehat{\Syrac}_n(\xi) := \mathbb{E}\,e^{-2\pi i \xi \Syrac(\mathbb{Z}/3^n\mathbb{Z}) / 3^n}$ for the characteristic function. Since $2$ is a primitive root modulo $3^n$ for every $n \ge 1$ (a classical result), every $\xi \in (\mathbb{Z}/3^n\mathbb{Z})^*$ can be written uniquely as $\xi = 2^k \bmod 3^n$ for some $k \in \{0, 1, \ldots, 2 \cdot 3^{n-1} - 1\}$. Throughout the paper we therefore work with the dual-side function
\begin{equation}
\psi_n(k) := \widehat{\Syrac}_n(2^k \bmod 3^n),
\end{equation}
which is periodic in $k$ with period $2 \cdot 3^{n-1}$.

Since $\Syrac$ is integer-valued, its characteristic function satisfies
\[
\widehat{\Syrac}_n(-\xi) = \overline{\widehat{\Syrac}_n(\xi)},
\]
where $\overline{\,\cdot\,}$ denotes complex conjugation. In particular, $|\widehat{\Syrac}_n(\cdot)|$ is invariant under $\xi \mapsto -\xi$. Using the classical fact
\[
2^{3^{n-1}} \equiv -1 \pmod{3^n},
\]
we have
\[
-2^k \equiv 2^{k+3^{n-1}} \pmod{3^n},
\]
and hence
\[
\psi_n(k+3^{n-1}) = \widehat{\Syrac}_n(-2^k) = \overline{\psi_n(k)}.
\]
Thus
\[
|\psi_n(k+3^{n-1})| = |\psi_n(k)|.
\]
We use this symmetry to reduce the search to representatives $0 \le k < 3^{n-1}$, and take $k^*(n)$ to be the smallest nonnegative representative on which $|\psi_n|$ attains its maximum.

\section{Numerical observations}\label{sec:observations}

For each $n$ in $1 \le n \le 21$, we compute $\psi_n$ on its full domain in IEEE 754 double-precision complex arithmetic and record the location of the maximum modulus. The data are summarised in Table~\ref{tab:main} and the underlying JSON file is included in the accompanying repository \cite{Repo}.

\begin{table}[h]
\centering
\caption{Numerical values of $M_n := \max_k |\psi_n(k)|$, the canonical maximiser $k^*(n)$, the floor formula prediction $\lfloor 4(n-1)/3 \rfloor$, and the relative gap to the second-largest value $|\psi_n|^2$.}
\label{tab:main}
\begin{tabular}{rllrr}
\toprule
$n$ & $M_n$ & $k^*(n)$ & $\lfloor 4(n-1)/3 \rfloor$ & $g_n / M_n^2$ \\
\midrule
1  & $5.77350 \times 10^{-1}$ & 0   & 0   & --- \\
2  & $3.77924 \times 10^{-1}$ & 2   & 1   & $6.54 \times 10^{-1}$ \\
3  & $2.52237 \times 10^{-1}$ & 3   & 2   & $4.06 \times 10^{-1}$ \\
4  & $1.76999 \times 10^{-1}$ & 4   & 4   & $4.21 \times 10^{-1}$ \\
5  & $1.29274 \times 10^{-1}$ & 5   & 5   & $3.94 \times 10^{-1}$ \\
6  & $9.61064 \times 10^{-2}$ & 6   & 6   & $1.06 \times 10^{-1}$ \\
7  & $7.58700 \times 10^{-2}$ & 8   & 8   & $1.05 \times 10^{-1}$ \\
8  & $6.08907 \times 10^{-2}$ & 9   & 9   & $2.20 \times 10^{-1}$ \\
9  & $4.80262 \times 10^{-2}$ & 10  & 10  & $1.19 \times 10^{-1}$ \\
10 & $3.82783 \times 10^{-2}$ & 12  & 12  & $2.75 \times 10^{-2}$ \\
11 & $3.19442 \times 10^{-2}$ & 13  & 13  & $1.33 \times 10^{-1}$ \\
12 & $2.64582 \times 10^{-2}$ & 14  & 14  & $1.11 \times 10^{-1}$ \\
13 & $2.20524 \times 10^{-2}$ & 16  & 16  & $2.52 \times 10^{-2}$ \\
14 & $1.91280 \times 10^{-2}$ & 17  & 17  & $1.39 \times 10^{-1}$ \\
15 & $1.62845 \times 10^{-2}$ & 18  & 18  & $1.36 \times 10^{-2}$ \\
16 & $1.44095 \times 10^{-2}$ & 20  & 20  & $1.09 \times 10^{-1}$ \\
17 & $1.25107 \times 10^{-2}$ & 21  & 21  & $2.27 \times 10^{-2}$ \\
18 & $1.11873 \times 10^{-2}$ & \textbf{23}  & 22  & $1.03 \times 10^{-1}$ \\
19 & $9.81571 \times 10^{-3}$ & 24  & 24  & $9.42 \times 10^{-3}$ \\
20 & $8.88459 \times 10^{-3}$ & \textbf{26}  & 25  & $1.10 \times 10^{-1}$ \\
21 & $7.87213 \times 10^{-3}$ & \textbf{28}  & 26  & $1.15 \times 10^{-2}$ \\
\bottomrule
\end{tabular}
\end{table}

In Table~\ref{tab:main}, $g_n := M_n^2 - |\psi_n(k_2^*)|^2$ where $k_2^*$ is the location of the second-largest value of $|\psi_n|^2$ over the canonical fundamental domain. The ratio $g_n / M_n^2$ measures how decisively $k^*(n)$ wins over the next candidate.

\begin{observation}[Partial agreement with the floor formula]\label{obs:floor}
For $n \in \{4, 5, \ldots, 17\}$ and for $n = 19$, $k^*(n)$ agrees with $\lfloor 4(n-1)/3 \rfloor$. The first deviation from the formula occurs at $n = 18$, where $k^*(18) = 23$ and $\lfloor 4 \cdot 17/3 \rfloor = 22$. Further deviations occur at $n = 20$ ($k^*(20) = 26$ versus $25$) and at $n = 21$ ($k^*(21) = 28$ versus $26$). Together with the small-$n$ values $n = 2, 3$ (which lie outside the long agreement window), the set of $n$ in $2 \le n \le 21$ for which $k^*(n) \ne \lfloor 4(n-1)/3 \rfloor$ is $\{2, 3, 18, 20, 21\}$. The floor formula thus describes $k^*(n)$ exactly on a long initial segment but is not a uniform identity in the present range.
\end{observation}

\begin{observation}[Slope near $4/3$]\label{obs:slope}
At $n = 21$,
\[
\frac{k^*(21)}{21} = \frac{28}{21} = \frac{4}{3} \quad \text{exactly}.
\]
More generally, $k^*(n)/n$ over $4 \le n \le 21$ lies in the interval $[1.333, 1.400]$ and is consistent with an approximately linear growth of slope near $4/3$. The interval slope
\[
\frac{k^*(21) - k^*(4)}{21 - 4} = \frac{28 - 4}{17} = \frac{24}{17} \approx 1.4118
\]
slightly exceeds $4/3$, the excess being a direct consequence of the endpoint deviation $k^*(21) - \lfloor 4(21-1)/3 \rfloor = 2$. (The interior deviations at $n = 18, 20$ do not affect the interval slope directly.)
\end{observation}

\begin{remark}[Finite-range decay fit]\label{rem:decay}
Over the finite range $9 \le n \le 21$, a power-law fit $M_n \sim C n^{-A}$ (least squares in $\log M_n$ versus $\log n$) yields $A \approx 2.10$. This is a finite-size observation only and should not be interpreted as an asymptotic claim. Tao's Proposition~\ref{prop:tao} gives super-polynomial decay, namely $M_n \ll_A n^{-A}$ for every fixed $A > 0$; if one defines an effective exponent over finite windows, Tao's result is compatible with such exponents increasing with $n$. The present data do not determine that behaviour.
\end{remark}

\begin{remark}[On the small-gap deviations]\label{rem:small-gap}
The deviations at $n = 18, 20, 21$ correlate with relatively small values of $g_n / M_n^2$: at $n = 18$ the ratio is $\approx 0.103$, at $n = 20$ it is $\approx 0.110$, and at $n = 21$ it is the smallest among $n \ge 13$, namely $\approx 0.012$. By contrast, at $n = 19$ where the floor formula is exact, the ratio is $\approx 0.0094$, which is smaller still. We do not draw structural conclusions from these correlations on the basis of $19$ data points alone.
\end{remark}

\begin{remark}[Tao's exponential-decay conjecture]\label{rem:tao-conjecture}
Tao remarks in \cite[Remark 1.15]{Tao2022} that the bound in his Proposition 1.17 should plausibly be improvable to $O(\exp(-c n))$ for some absolute $c > 0$. Our data in the range $n \le 21$ are best fit by a polynomial form $M_n \sim C n^{-2.1}$; an exponential fit over $9 \le n \le 21$ has roughly $20$ times larger sum-of-squared-residual in $\log M_n$. This is fully consistent with Proposition~\ref{prop:tao} but suggests that any transition to exponential decay, if it exists, must occur at scales beyond the present range.
\end{remark}

\section{Method}\label{sec:method}

The Syracuse random variable admits the explicit recursion of Tao \cite[Lemma 1.12]{Tao2022}. Writing $P_n(x) := \mathbb{P}(\Syrac(\mathbb{Z}/3^n\mathbb{Z}) = x)$, we have
\[
\Syrac(\mathbb{Z}/3^{n+1}\mathbb{Z}) \equiv 2^{-a} \, (3 \Syrac(\mathbb{Z}/3^n\mathbb{Z}) + 1) \pmod{3^{n+1}}, \quad a \sim \Geom(2),
\]
with $a$ independent of the level-$n$ variable. Taking expectations directly in the characteristic function yields the equivalent recursion in dual space:
\begin{equation}\label{eq:psi-recursion}
\psi_{n+1}(k) = \sum_{a=1}^{\infty} 2^{-a} \, \exp\!\left( - 2\pi i \cdot \frac{2^{k-a} \bmod 3^{n+1}}{3^{n+1}} \right) \, \psi_n\!\left( (k - a) \bmod 2 \cdot 3^{n-1} \right),
\end{equation}
where the inner exponential is a $3^{n+1}$-th root of unity at the integer $2^{k-a} \bmod 3^{n+1}$, well-defined for any integer $k-a$ since $\gcd(2, 3) = 1$. Equation \eqref{eq:psi-recursion} is the infinite-sum form: the geometric tail $\sum_{a=1}^\infty 2^{-a} = 1$ provides the correct normalisation directly.

We computed $\psi_n$ for $n = 1, \ldots, 21$ via two independent implementations:
\begin{enumerate}
\item Direct evaluation of \eqref{eq:psi-recursion} on the dual array of size $2 \cdot 3^{n-1}$. The sum over $a$ is truncated at $a = 80$, after which the geometric weight $2^{-a} < 10^{-24}$ is below double-precision noise.
\item Independent verification at small $n$ by direct evaluation of the level-$n+1$ recursion of $P_n$ on the support of size $3^n$, followed by discrete Fourier transform to obtain $\widehat{\Syrac}_n$, then comparison with $\psi_n$. Both implementations agree to relative error $< 10^{-10}$ at every $\xi$ and every $n \le 8$ (the range over which the independent verification is run).
\end{enumerate}

For $n \ge 17$, the dual array exceeds $1.7 \times 10^8$ complex entries and we computed in chunked passes. Up to $n = 19$, the computation runs on a laptop in a few hours. For $n = 20, 21$, we used a high-memory cloud instance with $961$ GB of RAM. At $n = 21$, the dual array has $2 \cdot 3^{20} \approx 7.0 \times 10^9$ entries and occupies $112$ GB as complex doubles; the auxiliary phase-factor array and the level-$22$ output array further dominate memory consumption. We did not compute $n = 22$: a single attempt during preparation of this paper failed because the working set exceeded available RAM during construction of the phase-factor array. Computing $n = 22$ in the present setup would require either a larger instance or restructured kernels.

A symmetry check is performed at each $n$: by the discussion in Section~\ref{sec:observations}, $|\psi_n(k)|$ must equal $|\psi_n(k + 3^{n-1})|$. We sample $1000$ values of $k$ and verify this equality; the maximum observed discrepancy is below $10^{-20}$ at every $n$, consistent with floating-point round-off.

Source code, the underlying JSON data for Table~\ref{tab:main}, and verification scripts are available at the repository \cite{Repo}.

\section{Discussion}\label{sec:discussion}

The most striking feature of the data is the following. The simple closed-form predictor $\lfloor 4(n-1)/3 \rfloor$ matches $k^*(n)$ exactly for $14$ consecutive values $n \in \{4, \ldots, 17\}$, suggesting at first inspection a structural identity. The agreement then fails at $n = 18, 20, 21$, with the deviations $+1, +1, +2$. Yet the ratio $k^*(21)/21$ equals $4/3$ exactly, so the underlying linear behaviour with slope $4/3$ remains consistent with the data even after the floor formula fails. This dual pattern --- exact agreement on a long initial segment, followed by deviations whose net effect leaves $k^*(21)/21$ at $4/3$ --- is, on its face, a numerical curiosity rather than a structural identity.

The deviations at $n \in \{18, 20, 21\}$ are notable for being clustered in the upper portion of our range and for the unexpected $+2$ value at $n = 21$. The finite-range fit $M_n \sim n^{-2.1}$ over $9 \le n \le 21$ recorded in Remark~\ref{rem:decay} is consistent with Tao's super-polynomial upper bound, which permits a slowly increasing effective exponent.

\subsection*{Open questions}

\begin{enumerate}
\item \textbf{Does $k^*(n)/n$ converge as $n \to \infty$, and is the limit $4/3$?} The available data are consistent with this, and at $n = 21$ we have $k^*(21)/21 = 4/3$ exactly, but a numerical observation on $18$ data points cannot distinguish $4/3$ from a nearby limit such as $24/17$.
\item Identify the structural reason, if any, for the deviations at $n \in \{18, 20, 21\}$. The pair $(18, 20)$ shares a deviation of $+1$ and even parity, while $21$ has odd parity and the unique deviation of $+2$; it is not clear whether these are coincidental, transient, or governed by a common arithmetic mechanism.
\item Determine whether the floor formula $\lfloor 4(n-1)/3 \rfloor$ is correct on density-one many $n$, density-zero many $n$, or something in between. Even with $14$ consecutive agreements, the deviations $\{18, 20, 21\}$ in the last four values of our range provide no clear picture.
\item Determine the precise asymptotic exponent of decay of $M_n$, and whether the transition predicted by \cite[Remark 1.15]{Tao2022} (to $\exp(-cn)$) occurs at scales accessible to direct computation.
\item Extend the computation to $n \ge 22$.
\end{enumerate}

\section*{Acknowledgements}

We thank Terence Tao for the foundational work in \cite{Tao2022, Tao2020} that motivates this paper.

\begin{thebibliography}{9}

\bibitem{Tao2022}
T.~Tao,
\emph{Almost all orbits of the Collatz map attain almost bounded values},
Forum Math. Pi \textbf{10} (2022), Paper No.~e12, 56 pp.
\href{https://doi.org/10.1017/fmp.2022.8}{doi:10.1017/fmp.2022.8}.
arXiv:1909.03562.

\bibitem{Tao2020}
T.~Tao,
\emph{Equidistribution of Syracuse random variables and density of Collatz preimages},
blog post (25 January 2020),
\url{https://terrytao.wordpress.com/2020/01/25/}.

\bibitem{Repo}
T.~Toguchi,
\emph{Reproducibility repository for the present paper},
\url{https://github.com/toguchitaku/syrac-extremal-fourier}.

\bibitem{Lagarias2010}
J.~C.~Lagarias (ed.),
\emph{The Ultimate Challenge: The $3x+1$ Problem},
American Mathematical Society, Providence, RI, 2010.

\end{thebibliography}

\end{document}