Irregular Stanley sequences plausibly do not have growth $\Theta(n^2/\log n)$

TL;DR

This paper provides numerical evidence suggesting that irregular Stanley sequences, specifically for n=4, do not grow at the conjectured rate of Θ(n^2/ log n), challenging previous assumptions about their growth behavior.

Contribution

The paper challenges the conjectured growth rate of irregular Stanley sequences for n=4 by providing numerical evidence that suggests a different growth pattern.

Findings

Numerical evidence against the Θ(n^2/ log n) growth rate for n=4.

Indications that the sequence's growth is closer to Ω(n^{2-δ}) for some δ > 0.

Discussion of limitations in the numerical methods used.

Abstract

Stanley sequences starting from the set $\{0, n\}$ where $n$ is a positive integer have long been conjectured to be divided into two types: the "regular" type where the growth rate is $\Theta(n^{\log_2(3)})$, and the "irregular" type where the growth rate is thought to be $\Theta(n^2/\log n)$. A paradigmatic case of a candidate irregular type is $n=4$, although to date no value of $n$ has been proven to have such a growth rate. Here, we provide strong numerical evidence against this conjectured growth rate for $n=4$. Specifically, for $n=4$, it seems plausible that the upper bound is $O(n^2/\log n)$ but that the lower bound is in fact $\Omega(n^{2-\delta})$ for some $\delta > 0$. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed.