Integer complexity and well-ordering

Abstract

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 ≥ 3 log 3 n for all n. Define the defect of n, denoted δ(n), to be n − 3 log 3 n. In this paper, we consider the set D := {δ(n) : n ≥ 1} of all defects. We show that as a subset of the real numbers, the set D is well-ordered, of order type ω ω . More specifically, for k ≥ 1 an integer, D ∩ [0, k) has order type ω k . We also consider some other sets related to D, and show that these too are well-ordered and have order type ω ω .

Integer complexity: Representing numbers of bounded defect

“…(9) is Proposition 3.1 from [1]. Also, although it will not be a focus of this paper, we will sometimes want to consider the set of all defects: Definition 2.2.…”

“…The paper [1] also defined the notion of a stable defect : Definition 2.3. We define a stable defect to be the defect of a stable number.…”

“…This sort of theorem is more powerful than it may appear; for instance, one can use it to show that the defect has unusual order-theoretic properties [1]: Theorem 1.4. (Defect well-ordering theorem) The set {δ(n) : n ∈ N}, considered as a subset of the real numbers, is well-ordered and has order type ω ω .…”

“…This is where a stronger version of Theorem 1.3 -Theorem 2.19 below, taken from [1] -is needed. This theorem ensures that, for any given r, not only is there a finite set S r of polynomials which represent all numbers with defect less than r by substituting in powers of 3; but in fact that it represents all such numbers "with the correct complexity".…”

“…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 ≥ 3 log 3 n for all n. Based on this, this author and Zelinsky defined [4] the "defect" of n, δ(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 [1]. 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 δ(n) < r, n has the form f (3 k 1 , .…”