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Melvyn B. Nathanson

Publications and source records attributed to Melvyn B. Nathanson.

At least 19 recordsLinked to original sources
arXiv2026

$B_h$-sets and perturbations in normed vector spaces

The subset $A = \{a_i:i \in I\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\varepsilon = \{\varepsilon_i:i \in I\}$ be a set of positive real numbers. An $\varepsilon$-perturbation of $A$ is a set $A' = \{a'_i:i\in I\}$ such that $|a_i'-a_i|<\varepsilon_i$ for all $i \in I$. Let $Δ_{hA} = \inf\{|x'-x| : x,x' \in hA \text{ and } x\neq x'\}$. It is proved that if $A$ is finite or countably infinite set with $Δ_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\varepsilon$-perturbation of $A$.

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arXiv2026

Sidon sets with $Δ$-separated sumsets in additive number theory

The set $A$ is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. A $B_2$-set is also called a Sidon set. A $B_{h,Δ}$-set is a $B_h$-set $A$ whose sumset $hA$ is $Δ$-separated, that is, $|x-x'| \geq Δ$ for all $x,x' \in hA$ with $x\neq x'$. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,Δ}$-sets contained in $\{1,2,\ldots, n\}$, that is, sets $A \subseteq \{1,2,\ldots, n\}$ such that, if $a,b,c,d \in A$ and $\{a,b\} \neq \{c,d\}$, then $|(a+b)-(c+d)| \geq Δ$.

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arXiv2026

A problem on sumset sizes of sets of lattice points

A central problem in additive number theory is to understand the set of sizes of $h$-fold sums of finite subsets of an additive abelian semigroup. It is proved that the "range of sumset sizes'' is the same for finite sets of integers and for finite sets of $n$-dimensional lattice points. The associated geometrical problem is to determine if these sumset size sets can be computed more efficiently with lattice points than with integers.

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arXiv2026

Problems in additive number theory, VII: The structure of additive $h$-bases for $n$

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new class of problems for these and related additive bases. The problems are designed, in part, to be susceptible to solution by AI.

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arXiv2026

Diversity, equity, and inclusion for problems in additive number theory

This is a survey of the diversity of problems in additive number theory. Equity requires the consideration of less currently popular problems, and suggests their inclusion in the additive canon. Of particular interest are problems about the sizes of sumsets of finite sets of integers and problems about the arithmetical structure of intersections of sumsets.

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arXiv2026

Problems and results on intersections of product sets and sumsets in semigroups

For every subset $A$ of a semigroup $S$, let $A^h$ be the set of all products of $h$ elements of $S$. If $(A)_{q\in Q}$ is a family of subsets of $S$, then $A = \bigcap_{q \in Q} A_q$ satisfies $A^h \subseteq \bigcap_{q \in Q} A_q^h$. The product intersection set $H(A_q) = \left\{h \in \mathbf{N}: A^h = \bigcap_{q \in Q} A_q^h \right\}$ is investigated.

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arXiv2026

Arithmetical structure of sumset intersections

The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = \{h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q\}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1,\ldots, h_0-1\} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1, h_0 \} \subseteq \mathcal{H}(A_q)$ but $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$.

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arXiv2026

Sumset size races for measurable sets

Let $G$ be a locally compact abelian group with Haar measure $μ$. For integers $n \geq 2$ and $H \geq 2$ and for any $n$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_H \in \mathbf{N}^n$, there exist measurable subsets $A_1,\ldots, A_n$ of $G$ such that the $n$-tuple $\left( μ(hA_1),\ldots, μ(hA_n) \right)$ has the same relative order as the $n$-tuple $\mathbf{u}_h$ for all $h = 1,\ldots, H$. For integers $m_{i,h}$ for $i =1,\ldots, n-1$ and $h = 1,\ldots, H$, there are Lebesgue measurable sets $A_1,\ldots, A_n$ in $\mathbf{R}$ such that $μ(hA_{i+1}) - μ(hA_i) = m_{i,h}$ for all $i$ and $h$.

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arXiv2025

Explicit sumset sizes in additive number theory

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $\mathcal{R}_{\mathbf{Z}}(h,k)= \{|hA|:A \subseteq {\mathbf{Z}} \text{ and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This paper constructs certain infinite families of finite sets of size $k$ and computes their $h$-fold sumset sizes.

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arXiv2025

Compression and complexity for sumset sizes in additive number theory

The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and $B_h$-sets). This paper considers the sets ${\mathcal R}_{\mathbf Z}(h,k)$ and ${\mathcal R}_{{\mathbf Z}^n}(h,k)$ of \emph{all} sizes of $h$-fold sums of sets of $k$ integers or of $k$ lattice points, and the geometric and computational complexity of the sets ${\mathcal R}_{\mathbf Z}(h,k)$ and ${\mathcal R}_{{\mathbf Z}^n}(h,k)$. For sumsets $hA$ with large diameter, there is a compression algorithm to construct sets $A'$ with $|hA'| = |hA|$ and small diameter.

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arXiv2025

Additive sumset sizes with tetrahedral differences

Experimental calculations suggest that the $h$-fold sumset sizes of 4-element sets of integers are concentrated at $h$ numbers that are differences of tetrahedral numbers. In this paper it is proved that these "popular" sumset sizes always exist. Explicit $h$-adically defined sets are constructed for each of these numbers.

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arXiv2025

Triangular and tetrahedral number differences of sumset sizes in additive number theory

The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range of sumset sizes of finite sets of integers and an unexpected pattern (related to the triangular and tetrahedral numbers) that appears in the distribution of popular sumset sizes of sets of size 4.

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arXiv2025

Addition theorems in partially ordered groups

Shnirel'man's inequality and Shnirel'man's basis theorem are fundamental results about sums of sets of positive integers in additive number theory. It is proved that these results are inherently order-theoretic and extend to partially ordered abelian and nonabelian groups. One abelian application is an addition theorem for sums of sets of $n$-dimensional lattice points.

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arXiv2025

$B_h$-sets of real and complex numbers

Let $K = \mathbb{R}$ or $\mathbb{C}$. An $n$-element subset $A$ of $K$ is a $B_h$-set if every element of $K$ has at most one representation as the sum of $h$ not necessarily distinct elements of $A$. Associated to the $B_h$ set $A = \{a_1,\ldots, a_n\}$ are the $B_h$-vectors $\mathbf{a} = (a_1,\ldots, a_n)$ in $K^n$. This paper proves that ``almost all'' $n$-element subsets of $K$ are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

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arXiv2025

Inverse problems for sumset sizes of finite sets of integers

Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $( |hA| )_{h=1}^{\infty}$ and $( |hA| )_{h=1}^{\infty}$.

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