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Melvyn B. Nathanson

Publications and source records attributed to Melvyn B. Nathanson.

At least 37 records · Page 2Linked to original sources
arXiv2024

Additive number theory and the Dyson transform

In 1942 Mann solved a famous problem, the $α+β$ conjecture, about the lower bound of the Shnirel'man density of sums of sets of positive integers. In 1945, Dyson generalized Mann's theorem and obtained a lower bound for the Shnirel'man density of rank $r$ sumsets. His proof introduced the Dyson transform, an important tool in additive number theory. This paper explains the background of Dyson's work, gives Dyson's proof of his theorem, and includes several applications of the Dyson transform, such as Kneser's inequality for sums of finite subsets of an arbitrary additive abelian group.

math.NT↗
arXiv2024

The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbf{R}^n$ with nonnegative coordinates. Permuting the coordinates, we can assume that $a_1 \geq \cdots \geq a_n$ and $b_1 \geq \cdots \geq b_n$. The vector $\mathbf{a}$ majorizes the vector $\mathbf{b}$, denoted $\mathbf{b} \preceq \mathbf{a}$, if $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$ and $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1,\ldots,n-1\}$. This paper proves theorems of Hardy-Littlewood-Pólya and Rado that $\mathbf{b} \preceq \mathbf{a}$ if and only if $P\mathbf{a} = \mathbf{b}$ for some doubly stochastic matrix $P$ if and only if $\mathbf{b}$ is in the $S_n$-permutohedron generated by $\mathbf{a}$.

math.CO↗
arXiv2024

Landau's converse to H\" older's inequality

By H\" older's inequality, if $\mathbf{x} \in \ell^p$, then $\mathbf{x}\mathbf{y} \in \ell^1$ for all $\mathbf{y} \in \ell^q$. Landau proved the converse result: If $\mathbf{x}\mathbf{y} \in \ell^1$ for all $\mathbf{y} \in \ell^q$, then $\mathbf{x} \in \ell^p$. This paper proves Landau's theorem and considers a related problem for Hilbert's inequality.

math.CA↗
arXiv2024

Increasing-decreasing patterns in the iteration of an arithmetic function

Let $Ω$ be a set of positive integers and let $f:Ω\rightarrow Ω$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m \in Ω$ has \textit{increasing-decreasing pattern} $V$ with respect to $f$ if, for all odd integers $i \in \{1,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) < f^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots < f^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for all even integers $i \in \{2,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) > f^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > f^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmetic function $f$ is \textit{wildly increasing-decreasing} if, for every finite sequence $V$ of positive integers, there exists an integer $m \in Ω$ such that $m$ has increasing-decreasing pattern $V$ with respect to $f$. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.

math.NT↗
arXiv2024

Finitely many implies infinitely many

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set $I$, then it is true for all of $I$. This paper describes some old and new results on infinite sets of linear and polynomial equations with the property that solutions for all finite subsets of the set of equations implies the existence of a solution for the infinite set of equations.

math.CO↗
arXiv2023

The fourth positive element in the greedy $B_h$-set

For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_r} \in A$ for all $r = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$-set. Then $a_1(h) = 1$, $a_2(h) = h+1$, and $a_3(h) = h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\left( h^3 + 3h^2 + 3h + 1\right) /2$ if $h$ is odd and $\left( h^3 + 2h^2 + 3h + 2\right) /2$ if $h$ is even.

math.NT↗
arXiv2023

The third positive element in the greedy $B_h$-set

For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_j} \in A$ for all $j = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$ set. One has $a_1(h) = 1$ and $a_2(h) = h+1$ for all $h$. Elementary proofs are given that $a_3(h) = h^2+h+1$ for all $h \geq 1$ and that $a_k(h) \leq \sum_{i=0}^{k-1} h^i$ for all $h \geq 1$ and $k \geq 1$.

math.NT↗
arXiv2023

Von Neumann-Steinhaus decomposition of parallelepipeds

A problem of Steinhaus was to partition a finite interval $I$ of the real line into countably infinitely many pairwise disjoint sets that are congruent in the sense that each set is a translate of a fixed set $A$. This paper describes von Neumann's solution of the Steinhaus problem and generalizes the result to parallelepipeds in $\mathbf{R}^n$.

math.CA↗
arXiv2023

Continuity of the roots of a polynomial

Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the polynomial.

math.RA↗
arXiv2023

Permutation Patterns of the Iterated Syracuse Function

Let $Ω$ be the set of odd positive integers and let $S:Ω\rightarrow Ω$ be the Syracuse function. It is proved that, for every permutation $σ$ of $(1,2,3)$, the set of triples of the form $(m,S(m),S^2(m))$ with permutation pattern $σ$ has positive density, and these densities are computed. However, there exist permutations $τ$ of $(1,2,3,4)$ such that no quadruple $(m,S(m), S^2(m), S^3(m))$ has permutation pattern $τ$. This implies the nonexistence of certain permutation patterns of $n$-tuples $(m,S(m),\ldots, S^{n-1}(m))$ for all $n \geq 4$.

math.NT↗
arXiv2023

Arithmetic functions and fixed points of powers of permutations

Let $σ$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( σ^k\right)$ count the number of fixed points of the $k$th power of $σ$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( σ^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $σ$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( σ^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations.

math.CO↗
arXiv2023

Polytopes, polyhedra, and the Farkas lemma

The Farkas lemma is proved and applied to obtain a structure theorem for polyhedra. These notes are based on a talk in the New York Number Theory Seminar on October, 20, 2022.

math.CO↗
arXiv2022

Interactions of zeros of of polynomials and multiplicity matrices

An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( μ_{i,j} )$ with rows enumerated by $i \in \{ 1,\ 2, \ldots, m \}$ and columns enumerated by $j \in \{ 0,1,\ldots, n \}$ whose coordinates are nonnegative integers satisfying the following two properties: (1) If $μ_{i,j} \geq 1$, then $j \leq n-1$ and $μ_{i,j+1} = μ_{i,j}-1$, and (2) the $j$th column sum of $M$ satisfies the inequality $\sum_{i=1}^{m} μ_{i, j} \leq n-j$ for all $j$. Let $K$ be a field of characteristic 0 and let $f(x)$ be a polynomial of degree $n$ with coefficients in $K$. Let $f^{(j)}(x)$ be the $j$th derivative of $f(x)$. Let $Λ= ( λ_1,\ldots, λ_{m})$ be a sequence of distinct elements of $K$. For $i \in \{1, 2, \ldots, m \}$ and $j \in \{1,2,\ldots, n\}$, let $ μ_{i,j}$ be the multiplicity of $λ_i$ as a zero of the polynomial $f^{(j)}(x)$. The $m \times (n+1)$ matrix $M_f(Λ) = ( μ_{i,j} )$ is called the multiplicity matrix of the polynomial $f(x)$ with respect to $Λ$. An open problem is to classify the multiplicity matrices that are multiplicity matrices of polynomials in $K[x]$ and to construct multiplicity matrices that are not multiplicity matrices of polynomials.

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