arXiv subjects

Harry Altman

Publications and source records attributed to Harry Altman.

11 recordsLinked to original sources

arXiv2026

A Ring structure on the Class of Combinatorial Games

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games form a Group. However, Conway product is not well defined on equivalence classes of arbitrary games (though it is well defined for surreals). We consider an equivalence relation finer than Conway's and show that under such a relation combinatorial games actually form a Ring. We hint to other possible relations on the Class of combinatorial games.

math.CO↗

arXiv2024

Bounding finite-image sequences of length $\omega^k$

Given a well-quasi-order $X$ and an ordinal $\alpha$, the set $s^F_\alpha(X)$ of transfinite sequences on $X$ with length less than $\alpha$ and with finite image is also a well-quasi-order, as proven by Nash-Williams. Before Nash-Williams proved it for general $\alpha$, however, it was proven for $\alpha<\omega^\omega$ by Erd\H{o}s and Rado. In this paper, we revisit Erd\H{o}s and Rado's proof and improve upon it, using it to obtain upper bounds on the maximum linearization of $s^F_{\omega^k}(X)$ in terms of $k$ and $o(X)$, where $o(X)$ denotes the maximum linearization of $X$. We show that, for fixed $k$, $o(s^F_{\omega^k}(X))$ is bounded above by a function which can roughly be described as $(k+1)$-times exponential in $o(X)$. We also show that, for $k\le 2$, this bound is not far from tight.

math.LO↗

arXiv2021

Integer complexity: Stability and self-similarity

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.

math.NT↗

arXiv2019

Maximum linearizations of lower sets in $\mathbb{N}^m$ with application to monomial ideals

We compute the type (maximum linearization) of the well partial order of bounded lower sets in $\mathbb{N}^m$, ordered under inclusion, and find it is $\omega^{\omega^{m-1}}$. Moreover we compute the type of the set of all lower sets in $\mathbb{N}^m$, a topic studied by Aschenbrenner and Pong, and find that it is equal to \[ \omega^{\sum_{k=1}^{m} \omega^{m-k}\binom{m}{k-1} }+ 1. \] As a consequence we deduce corresponding bounds on effectively given sequences of monomial ideals in $F[X,Y]$ where $F$ is a field.

math.LO↗

arXiv2018

Integer complexity: The integer defect

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 $n$, leading this author and Zelinsky to define the defect of $n$, $\delta(n)$, to be the difference $\|n\|-3\log_3 n$. Meanwhile, in the study of addition chains, it is common to consider $s(n)$, the number of small steps of $n$, defined as $\ell(n)-\lfloor\log_2 n\rfloor$, an integer quantity. So here we analogously define $D(n)$, the integer defect of $n$, an integer version of $\delta(n)$ analogous to $s(n)$. Note that $D(n)$ is not the same as $\lceil \delta(n) \rceil$. We show that $D(n)$ has additional meaning in terms of the defect well-ordering considered in [3], in that $D(n)$ indicates which powers of $\omega$ the quantity $\delta(n)$ lies between when one restricts to $n$ with $\|n\|$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D(n)\le 1$, and use this to generalize a result of Rawsthorne [18].

math.NT↗

arXiv2016

Integer complexity: algorithms and computational results

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. Define $n$ to be stable if for all $k\ge 0$, we have $\|3^k n\|=\|n\|+3k$. In [7], this author and Zelinsky showed that for any $n$, there exists some $K=K(n)$ such that $3^K n$ is stable; however, the proof there provided no upper bound on $K(n)$ or any way of computing it. In this paper, we describe an algorithm for computing $K(n)$, and thereby also show that the set of stable numbers is a computable set. The algorithm is based on considering the defect of a number, defined by $\delta(n):=\|n\|-3\log_3 n$, building on the methods presented in [3]. As a side benefit, this algorithm also happens to allow fast evaluation of the complexities of powers of $2$; we use it to verify that $\|2^k 3^\ell\|=2k+3\ell$ for $k\le48$ and arbitrary $\ell$ (excluding the case $k=\ell=0$), providing more evidence for the conjecture that $\|2^k 3^\ell\|=2k+3\ell$ whenever $k$ and $\ell$ are not both zero. An implementation of these algorithms in Haskell is available.

math.NT↗

arXiv2016

Integer Complexity: Representing Numbers of Bounded Defect

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.

math.NT↗

arXiv2015

Intermediate arithmetic operations on ordinal numbers

There are two well-known ways of doing arithmetic with ordinal numbers: the "ordinary" addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the "natural" (or Hessenberg) addition and multiplication (denoted $\oplus$ and $\otimes$), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted $\times$), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote $\alpha^{\times\beta}$. (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we will denote this $\alpha^{\otimes\beta}$. We show that $\alpha^{\otimes(\beta\oplus\gamma)} = (\alpha^{\otimes\beta}) \otimes(\alpha^{\otimes\gamma})$ and that $\alpha^{\otimes(\beta\times\gamma)}=(\alpha^{\otimes\beta})^{\otimes\gamma}$; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a "natural exponentiation" satisfying reasonable algebraic laws.

math.LO↗

arXiv2014

Internal Structure of Addition Chains: Well-Ordering

An addition chain for $n$ is defined to be a sequence $(a_0,a_1,\ldots,a_r)$ such that $a_0=1$, $a_r=n$, and, for any $1\le k\le r$, there exist $0\le i, j<k$ such that $a_k = a_i + a_j$; the number $r$ is called the length of the addition chain. The shortest length among addition chains for $n$, called the addition chain length of $n$, is denoted $\ell(n)$. The number $\ell(n)$ is always at least $\log_2 n$; in this paper we consider the difference $\delta^\ell(n):=\ell(n)-\log_2 n$, which we call the addition chain defect. First we use this notion to show that for any $n$, there exists $K$ such that for any $k\ge K$, we have $\ell(2^k n)=\ell(2^K n)+(k-K)$. The main result is that the set of values of $\delta^\ell$ is a well-ordered subset of $[0,\infty)$, with order type $\omega^\omega$. The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.

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arXiv2013

Integer Complexity and Well-Ordering

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 $n$. Define the defect of $n$, denoted $\delta(n)$, to be $\|n\| - 3\log_3 n$. In this paper, we consider the set $\mathscr{D} := \{\delta(n): n \ge 1 \}$ of all defects. We show that as a subset of the real numbers, the set $\mathscr{D}$ is well-ordered, of order type $\omega^\omega$. More specifically, for $k\ge 1$ an integer, $\mathscr{D}\cap[0,k)$ has order type $\omega^k$. We also consider some other sets related to $\mathscr{D}$, and show that these too are well-ordered and have order type $\omega^\omega$.

math.NT↗

arXiv2012

Numbers with Integer Complexity Close to the Lower Bound

Define $|n|$ to be the complexity of $n$, the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $|n|\ge 3\log_3 n$ for all $n$. Define the defect of $n$, denoted $\delta(n)$, to be $|n|-3\log_3 n$; in this paper we present a method for classifying all $n$ with $\delta(n)<r$ for a given $r$. From this, we derive several consequences. We prove that $|2^m 3^k|=2m+3k$ for $m\le 21$ with $m$ and $k$ not both zero, and present a method that can, with more computation, potentially prove the same for larger $m$. Furthermore, defining $A_r(x)$ to be the number of $n$ with $\delta(n)<r$ and $n\le x$, we prove that $A_r(x)=\Theta_r((\log x)^{\lfloor r \rfloor+1})$, allowing us to conclude that the values of $|n|-3\log_3 n$ can be arbitrarily large.

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