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Monomial Congruences.

Melvyn B. Nathanson — 1978

Monatshefte für Mathematik

Connected Components of Arithmetic Graphs.

Melvyn B. Nathanson — 1980

Monatshefte für Mathematik

Unique representation bases for the integers

Melvyn B. Nathanson — 2003

Acta Arithmetica

Problems in additive number theory, II: Linear forms and complementing sets

Melvyn B. Nathanson — 2009

Journal de Théorie des Nombres de Bordeaux

Let $ϕ (x_{1}, ..., x_{h}, y) = u_{1} x_{1} + \dots + u_{h} x_{h} + v y$ be a linear form with nonzero integer coefficients $u_{1}, ..., u_{h}, v .$ Let $𝒜 = (A_{1}, ..., A_{h})$ be an $h$ -tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $ϕ$ and the sets $𝒜$ and $B$ as follows : $R_{𝒜, B}^{(ϕ)} (n) = card (\begin{matrix} {(a_{1}, ..., a_{h}, b) \in A_{1} & \times \dots \times A_{h} \times B : \\ ϕ (a_{1}, ..., a_{h}, b) = n} \end{matrix}) .$ If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set ${ϕ (a_{1}, ..., a_{h}, 0) : (a_{1}, ..., a_{h}) \in A_{1} \times \dots \times A_{h}} .$ Other results for complementing sets with respect to linear forms are also proved....

Partititions of the Natural Numbers into Infinitely Oscillating Bases and Nonbases

Paul Erdös; Melvyn B. Nathanson — 1976

Commentarii mathematici Helvetici

Polynomial growth of sumsets in abelian semigroups

Melvyn B. Nathanson; Imre Z. Ruzsa — 2002

Journal de théorie des nombres de Bordeaux

Let $S$ be an abelian semigroup, and $A$ a finite subset of $S$ . The sumset $h A$ consists of all sums of $h$ elements of $A$ , with repetitions allowed. Let $| h A |$ denote the cardinality of $h A$ . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial $p (t)$ such that $| h A | = p (h)$ for all sufficiently large $h$ . Lattice point counting is also used to prove that sumsets of the form $h_{1} A_{1} + \dots + h_{r} A_{r}$ have multivariate polynomial growth.

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson; Kevin O'Bryant — 2015

Acta Arithmetica

A geometric progression of length k and integer ratio is a set of numbers of the form $a, a r, . . ., a r^{k - 1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${(a_{i})}_{i = 1}^{\infty}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃_{i = 1}^{\infty} (a_{2 i}, a_{2 i - 1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...