Integer complexity: Stability and self-similarity

TL;DR

This work advances the theory of integer complexity by revealing a self-similar structure in the defect set, showing that the closure $\\overline{\\mathscr{D}}$ satisfies $\\overline{\\mathscr{D}}' = \\\overline{\\mathscr{D}} + 1$ and that defects arising from substantial low-defect polynomials typically attain their naïve upper bounds. The authors develop the framework of substantial low-defect polynomials, augmented expressions, and good coverings to efficiently represent numbers with small defects and to analyze the asymptotics of complexity for families like $b(a3^k+1)3^\\ell$. The main contributions include a detailed structural decomposition of $\\overline{\\mathscr{D}}$, proofs of self-similarity at all scales, and several degree-1 results that resolve conjectures of Arias (with computable bounds). These results clarify stability phenomena in integer complexity and provide algorithmic techniques for certifying when the naïve upper bounds are achieved, significantly advancing understanding of the defect landscape and its applications to conjectures and variants.

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 $δ(n):=||n||-3\log_3 n$, is known to be a well-ordered subset of $[0,\infty)$, with order type $ω^ω$. 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 $δ(n)\le s$ 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 $ω^ω$, 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 naïve 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 naïve upper bound. These results resolve various earlier conjectures of the second author.

Theorems & Definitions (136)

• Definition 1.1
• Theorem 1.2: paperwo
• Theorem 1.3: Self-similarity theorem
• Theorem 1.5: Combined index-value shift relation
• Theorem 1.6: Split index-value shift relation
• Theorem 1.7: Defect closure theorem
• Theorem 1.9: Expression defect theorem
• Definition 1.10
• Theorem 1.11: Stability theorem; paper1
• Definition 1.12