Integer Complexity: Representing Numbers of Bounded Defect

TL;DR

This paper refines the representation of numbers with bounded defect in integer complexity, ensuring precise characterization by modifying low-defect polynomials to exclude extraneous numbers.

Contribution

It introduces a method to truncate low-defect polynomials, accurately representing numbers with defect less than a fixed real number, eliminating extraneous representations.

Findings

Successfully modifies polynomial sets to precisely represent numbers with bounded defect.

Removes extraneous numbers from polynomial representations by truncating polynomials.

Provides a refined framework for understanding integer complexity and defect sets.

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. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$. Based on this, this author and Zelinsky defined the "defect" of $n$, $\delta(n):=\|n\|-3\log_3 n$, and this author showed that the set of all defects is a well-ordered subset of the real numbers. This was accomplished by showing that for a fixed real number $r$, there is a finite set $S$ of polynomials called "low-defect polynomials" such that for any $n$ with $\delta(n)<r$, $n$ has the form $f(3^{k_1},\ldots,3^{k_r})3^{k_{r+1}}$ for some $f\in S$. However, using the polynomials produced by this method, many extraneous $n$ with $\delta(n)\ge r$ would also be represented. In this paper we show how to remedy this and modify $S$ so as to represent precisely the $n$ with $\delta(n)<r$ and remove anything extraneous. Since the same polynomial can represent both $n$ with $\delta(n)<r$ and $n$ with $\delta(n)\ge r$, this is not a matter of simply excising the appropriate polynomials, but requires "truncating" the polynomials to form new ones.