Ternary Digits of Powers of Two

TL;DR

The paper investigates whether the ternary digits of powers of two are uniformly distributed in the limit and whether the base-3 expansion of $\log_3 2$ is normal. It combines definitions of aggregate digit counts with extensive computational experiments up to $n\le 10^6$, revealing strong evidence for uniform aggregate frequencies ($1/3$ per digit and $3^{-k}$ for blocks of length $k$) and Benford-type leading-digit biases. A very special case of Baker's Theorem is applied to bound patterns after the leading digit, constraining possible run lengths of zeros. The work also analyzes $\alpha=\log_3 2$, providing computational support that this constant is normal to base 3, while highlighting that such normality remains unproved and does not directly follow from the uniformity observed in $2^n$. Overall, the findings lend substantial numerical support to uniform distribution phenomena in these deterministic sequences and connect them to foundational results in uniform distribution and normality.

Abstract

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same length, as the exponent $n$ approaches infinity (\textit{Uniform Distribution in the limit}). Currently the answer to this question is unknown. Even a much weaker conjecture by Erdös is still open. But we present computational results (up to $n = 10^6$) supporting uniform distribution in the limit. In this context, we discuss implications of Benford's Law and a special case of Baker's Theorem. Then we investigate the infinite sequence of ternary digits of $\log_3(2)$. There are analogous questions about the distribution of strings of 0s, 1s, and 2s in that sequence. If there is uniform distribution in the limit, then $\log_3(2)$ is called \textit{normal to base 3}. In the absence of definitive results, we can offer again computational evidence from the first $10^6$ ternary digits of $\log_3(2)$, strongly supporting the conjecture that $\log_3(2)$ is normal to base 3.

Figures (2)

• Figure 1: Deviation of aggregate digit frequencies $F_d(N)$ from $1/3$ for exponents $N\le 2000$. Colors: blue $=0$, orange $=1$, green $=2$.
• Figure 2: Deviation of the frequency $f_d(n)$ of each digit in individual powers $2^n$ from the uniform value $1/3$ for $1\le n\le2000$. Colors are consistent with Figure \ref{['fig:agg']}: blue $=0$, orange $=1$, green $=2$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Theorem 1: Benford's Law for ternary digits
• proof
• Theorem 2: Average count in leading digits
• proof
• Theorem 3: Uniform distribution of frequency in leading digits
• proof
• Remark 1
• Theorem 4: Baker 1975 -- very special case