On the minimal length of addition chains

TL;DR

This paper studies the distribution of the minimal addition-chain length by introducing $F(m,r)$, the count of integers in $[2^m,2^{m+1})$ with $\ell(n)\le m+r$. It provides precise upper and lower bounds for $F(m,\tfrac{cm}{\log m})$ in terms of $cm$ and subpolynomial corrections, extending Erdős’s result on $\ell(n)$ for almost all $n$. The authors develop a detailed combinatorial framework that partitions addition steps into doubling, large, midsize, and small types, analyzes their interdependencies via block structures, and leverages marked-block arguments to bound the number of candidate chains. A constructive lower-bound argument using binary expansion yields matching exponential-scale growth up to logarithmic corrections. Together, these results push toward a fuller understanding of how the minimal-length addition chains are distributed across integers, while highlighting remaining gaps between upper and lower estimates.

Abstract

We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erdős which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$

Theorems & Definitions (41)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 3.1
• Theorem 3.2
• Definition 4.1
• Definition 4.2
• Lemma 4.1
• proof
• Lemma 4.2