arXiv subjects
Publications and source records attributed to Melvyn B. Nathanson.
An exponential automorphism of $\mathbf{C}$ is a function $α: \mathbf{C} \rightarrow \mathbf{C}$ such that $α(z_1 + z_2) = α(z_1) + α(z_2)$ and $α\left( e^z \right) = e^{α(z)}$ for all $z, z_1, z_2 \in \mathbf{C}$. Jan Mycielski asked if $α(\ln 2) = \ln 2$ and if $α(2^{1/k}) = 2^{1/k}$ for $k = 2, 3, 4$ and for all exponential automorphisms $α$. These questions are answered modulo a multiple of $2πi$ and a root of unity.
Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F} = \{f_λ :λ\in Λ\}$ is a set of polynomials in $\mathbf{C}[t_1,\ldots, t_n]$ and if $^*\mathcal{G} = \{g_λ :λ\in Λ\}$ is a set of polynomials in $^*\mathbf{C}_0[t_1,\ldots, t_n]$ such that $g_λ$ is an infinitesimal deformation of $f_λ$ for all $λ\in Λ$, then the nonstandard affine variety $^*V_0(\mathcal{G})$ is an infinitesimal deformation of the affine variety $V(\mathcal{F})$.
Menon's identity is $\sum_{a \in A}^m (a-1,m) = d(m) φ(m)$, where $A$ is a reduced set of residues modulo $m$. This paper contains elementary proofs of some generalizations of this result.
This is an exposition of Gauss's proof of Descartes's rule of signs.
A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ not necessarily distinct elements of $A$. The asymptotic basis $A$ is minimal if removing any element of $A$ destroys every representation of infinitely many integers, and so $A\setminus \{a\}$ is not an asymptotic basis of order $h$ for all $a\in A$. In this paper, a new class of minimal asymptotic bases is constructed.
Here is a direct problem for Sidon sets: Given a linear form $φ= c_1 x_1 + \cdots + c_h x_h $, construct and describe sets $A$ that are Sidon sets for $φ$. This paper considers an inverse problem for Sidon sets: Given a set $A$, determine the linear forms $φ$ such that $A$ is a Sidon set for $φ$.
An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $θ\in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < θ$. A greedy algorithm constructs an $n$-term underapproximation of $θ$. For some but not all numbers $θ$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.
Let $φ(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h $ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $φ$-Sidon set if, for all $h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $ (a'_1,\ldots, a'_h) \in A^h$, the relation $φ(a_1,\ldots, a_h) = φ(a'_1,\ldots, a'_h)$ implies $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for the linear form $φ$ if and only if the set of coefficients of $φ$ has distinct subset sums. In a normed vector space with $φ$-Sidon sets, every infinite sequence of vectors is asymptotic to a $φ$-Sidon set of vectors. Results on $p$-adic perturbations of $φ$-Sidon sets of integers and bounds on the growth of $φ$-Sidon sets of integers are also obtained.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both convergent and divergent harmonic series.
Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal{B} }(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$. It is proved that $\liminf_{n\rightarrow \infty} g_{\mathcal{B} }(n) \geq 2$ implies $\limsup_{n\rightarrow \infty} g_{\mathcal{B} }(n) = \infty$.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent harmonic series.
A subset $A$ of an additive abelian group is an $h$-Sidon set if every element in the $h$-fold sumset $hA$ has a unique representation as the sum of $h$ not necessarily distinct elements of $A$. Let $\mathbf{F}$ be a field of characteristic 0 with a nontrivial absolute value, and let $A = \{a_i :i \in \mathbf{N} \}$ and $B = \{b_i :i \in \mathbf{N} \}$ be subsets of $\mathbf{F}$. Let $\varepsilon = \{ \varepsilon_i:i \in \mathbf{N} \}$, where $\varepsilon_i > 0$ for all $i \in \mathbf{N}$. The set $B$ is an $\varepsilon$-perturbation of $A$ if $|b_i-a_i| < \varepsilon_i$ for all $i \in \mathbf{N}$. It is proved that, for every $\varepsilon = \{ \varepsilon_i:i \in \mathbf{N} \}$ with $\varepsilon_i > 0$, every set $A = \{a_i :i \in \mathbf{N} \}$ has an $\varepsilon$-perturbation $B$ that is an $h$-Sidon set. This result extends to sets of vectors in $\mathbf{F}^n$.
A polynomial is real-rooted if all of its roots are real. For every polynomial $f(t) \in {\mathbf R}[t]$, the Hermite-Sylvester theorem associates a quadratic form $Φ_2$ such that $f(t)$ is real-rooted if and only if $Φ_2$ is positive semidefinite. In this note, for every positive integer $m$, an $2m$-adic form $Φ_{2m}$ is constructed such that $f(t)$ is real-rooted if and only if $Φ_{2m}$ is positive semidefinite for some $m$ if and only if $Φ_{2m}(x_1,\ldots, x_n)$ is positive semidefinite for all $m$.
For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $σ_c < 1$. The number $σ_c$ is computed. This generalizes and strengthens a classical theorem of Kempner on the convergence of the sum of the reciprocals of a sequence of integers with missing decimal digits.
Let $\mathbf{A} = (A_1,\ldots, A_q)$ be a $q$-tuple of finite sets of integers. Associated to every $q$-tuple of nonnegative integers $\mathbf{h} = (h_1,\ldots, h_q)$ is the linear form $\mathbf{h}\cdot \mathbf{A} = h_1 A_1 + \cdots + h_qA_q$. The set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ consists of all elements of this sumset with at least $t$ representations. The structure of the set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ is computed for all sufficiently large $h_i$.
Epistemological and sociological questions about ownership of mathematical and scientific discoveries.
Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.
An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.