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 D of defects, differences δ(n) := n − 3 log 3 n, is known to be a well-ordered subset of [0, ∞), with order type ω ω . This is proved by showing that, for any r, there is a finite set Ss of certain multilinear polynomials, called low-defect polynomials, such that δ(n) ≤ s if and only if one can write n = f (3 k 1 , . . . , 3 kr )3 k r+1 . [3,4] In this paper we show that, in addition to it being true that D (and thus D) has order type ω ω , it satisifies a sort of self-similarity property with DThis 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 ℓ will, for all sufficiently large k, have complexity equal to the naïve upper bound. These results resolve various earlier conjectures of the second author. [11]