[Paper Review] Benford's law for the $3x+1$ function

Jeffrey C. Lagarias, K. Soundararajan|ArXiv.org|Sep 7, 2005

Benford’s Law and Fraud Detection12 references4 citations

TL;DR

This paper demonstrates that for most initial values in the $3x+1$ function, the first $N$ iterates approximately satisfy Benford’s law in any base $B \geq 2$, including powers of 2. Using a stochastic model of the $3x+1$ dynamics and discrepancy theory, it proves that the logarithmic sequences $\{\log_B x_k\}$ are uniformly distributed modulo one for most starting values, with the discrepancy decaying as $O(N^{-1/36})$, establishing a strong quantitative form of Benford’s law for the $3x+1$ iterates.

ABSTRACT

We show that for most choices of an initial seed $x_0$, the sequence of the first $N$ iterates of $x_0$ under the $3x+1$ map approximately satisfies Benford's law.

Motivation & Objective

• To investigate whether the initial iterates of the $3x+1$ function approximately satisfy Benford’s law in base $B$ for large $N$.
• To quantify the proportion of initial values $x_0$ for which the sequence $\{\log_B T^{(k)}(x_0)\}$ is uniformly distributed modulo one.
• To extend previous results on Benford’s law for $3x+1$ to include all bases $B \geq 2$, including powers of 2, which were previously anomalous in prior work.
• To provide a rigorous, quantitative bound on the discrepancy of the logarithmic sequences derived from $3x+1$ iterates.

Proposed method

• Model the $3x+1$ dynamics using a stochastic process where each step independently applies $T(n) = \frac{3n}{2}$ or $T(n) = \frac{n}{2}$ with equal probability, corresponding to $\log_B x_k$ increasing by $\theta_1 = \log_B(3/2)$ or $\theta_2 = \log_B(1/2)$.
• Define the discrepancy $D(\{y_1, \dots, y_N\})$ of the sequence $y_k = \log_B x_k$ as a measure of deviation from uniform distribution modulo one.
• Use the fact that the discrepancy of the stochastic process $\mathcal{P}(\theta_1, \theta_2, y_0 = 0)$ is bounded in expectation via a result on Diophantine approximation with exponent $\alpha = 7.616$.
• Apply Markov’s inequality to bound the probability that the discrepancy exceeds $N^{-1/36}$, yielding a quantitative upper bound on the exceptional set of initial values.
• Establish a connection between the actual $3x+1$ iterates and a truncated version of the stochastic process via a lifting argument, showing that discrepancies are preserved for most initial values.
• Use periodicity and interval covering arguments to extend the result from $X = 2^N$ to general $X > 2^N$.

Experimental results

Research questions

• RQ1 Do the initial $N$ iterates of the $3x+1$ function approximately satisfy Benford’s law in base $B$ for most starting values?
• RQ2 Can a quantitative bound be established on the discrepancy of $\{\log_B x_k\}$ modulo one for the $3x+1$ iterates?
• RQ3 Does the result hold for all bases $B \geq 2$, including powers of 2, despite previous anomalies in odd-iterate-only models?
• RQ4 How does the discrepancy decay as $N$ increases, and what fraction of initial values are 'exceptional' in violating Benford’s law?

Key findings

• For most initial values $x_0$ in $1 \leq x_0 \leq X$ with $X \geq 2^N$, the sequence $\{\log_B x_k\}_{k=1}^N$ has discrepancy $D \leq N^{-1/36} + 2^{2 - \frac{1}{100}N}$.
• The exceptional set of initial values $x_0$ for which the discrepancy exceeds $N^{-1/36}$ has cardinality at most $C N^{-1/36} 2^N + 2^{1 + \frac{99}{100}N}$ for some constant $C$.
• The discrepancy bound holds uniformly for all bases $B \geq 2$, including when $B$ is a power of 2, resolving a previous anomaly in odd-iterate-only models.
• The expected discrepancy of the stochastic process modeling the $3x+1$ iterates is bounded by $O(N^{-1/18})$, which implies the main result via Markov’s inequality.
• The result is established by showing that the discrepancy of the actual $3x+1$ iterates is very close to that of a related stochastic process, which is well-understood via Diophantine approximation.