Integer complexity: Stability and self-similarity

Abstract

Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known to be a well-ordered subset of $[0,\infty)$, with order type $\omega^\omega$. This is proved by showing that, for any $r$, there is a finite set $\mathcal{S}_s$ of certain multilinear polynomials, called low-defect polynomials, such that $\delta(n)\le s$ if and only if one can write $n = f(3^{k_1},\ldots,3^{k_r})3^{k_{r+1}}$. In this paper we show that, in addition to it being true that $\mathscr{D}$ (and thus $\overline{\mathscr{D}}$) has order type $\omega^\omega$, this set satisifies a sort of self-similarity property with $\overline{\mathscr{D}}' = \overline{\mathscr{D}} + 1$. This is proven by restricting attention to substantial low-defect polynomials, ones that can be themselves written efficiently in a certain sense, and showing that in a certain sense the values of these polynomials at powers of $3$ have complexity equal to the na\"ive upper bound most of the time. As a result, we also prove that, under appropriate conditions on $a$ and $b$, numbers of the form $b(a3^k+1)3^\ell$ will, for all sufficiently large $k$, have complexity equal to the na\"ive upper bound. These results resolve various earlier conjectures of the second author.