arXiv subjects

Melvyn B. Nathanson

Publications and source records attributed to Melvyn B. Nathanson.

At least 145 records · Page 8Linked to original sources

arXiv2006

Linear quantum addition rules

The quantum integer $[n]_q$ is the polynomial $1 + q + q^2 + ... + q^{n-1}.$ Two sequences of polynomials $\mathcal{U} = \{u_n(q)\}_{n=1}^{\infty}$ and $\mathcal{V} = \{v_n(q)\}_{n=1}^{\infty}$ define a {\em linear addition rule} $\oplus$ on a sequence $\mathcal{F} = \{f_n(q)\}_{n=1}^{\infty}$ by $f_m(q)\oplus f_n(q) = u_n(q)f_m(q) + v_m(q)f_n(q).$ This is called a {\em quantum addition rule} if $[m]_q \oplus [n]_q = [m+n]_q$ for all positive integers $m$ and $n$. In this paper all linear quantum addition rules are determined, and all solutions of the corresponding functional equations $f_m(q)\oplus f_n(q) = f_{m+n}(q)$ are computed.

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arXiv2006

Decomposing sequences into monotonic subsequences

The function f:X -> Y is called k-monotonically increasing if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is k-monotonically increasing if and only if every set of k+1 positive integers contains two integers x,x' with x < x' such that f(x) <= f(x'). The function f:X \to Y is called k-monotonic if there is a partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically increasing or monotonically decreasing for i=1,...,k. It is also proved that there does not exist a k-monotonic function from N onto Q.

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arXiv2006

The Caccetta-Haggkvist conjecture and additive number theory

The Caccetta-Haggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used methods and results from additive number theory to prove the conjecture for Cayley graphs and for vertex-transitive graphs. This expository paper contains a survey of results on the Caccetta-Haggkvist conjecture, and complete proofs of the conjecture in the case of Cayley and vertex-transitive graphs.

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arXiv2005

Maximal Sidon Sets and Matroids

Let X be a subset of an abelian group and a_1,...,a_h,a'_1,...,a'_h a sequence of 2h elements of X such that a_1 + ... + a_h = a'_1 + ... + a'_h. The set X is a Sidon set of order h if, after renumbering, a_i = a'_i for i = 1,..., h. For k \leq h, the set X is a generalized Sidon set of order (h,k), if, after renumbering, a_i = a'_i for i = 1,..., k. It is proved that if X is a generalized Sidon set of order (2h-1,h-1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.

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arXiv2005

A new upper bound for finite additive bases

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this paper it is proved that limsup_{k\to\infty} n(2,k)/k^2 \leq 0.4789.

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arXiv2005

Quadratic addition rules for quantum integers

For every positive integer $n$, the quantum integer $[n]_q$ is the polynomial $[n]_q = 1 + q + q^2 + ... + q^{n-1}.$ A quadratic addition rule for quantum integers consists of sequences of polynomials $\mathcal{R}' = \{r'_n(q)\}_{n=1}^{\infty}$, $\mathcal{S}' = \{s'_n(q)\}_{n=1}^{\infty}$, and $\mathcal{T}' = \{t'_{m,n}(q)\}_{m,n=1}^{\infty}$ such that $[m+n]_q = r'_n(q)[m]_q + s'_m(q)[n]_q + t'_{m,n}(q)[m]_q[n]_q$ for all $m$ and $n.$ This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials \polf that satisfy the associated functional equation $f_{m+n}(q)= r'_n(q)f_m(q) + s'_m(q)f_n(q) + t'_{m,n}f_m(q)f_n(q).$

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arXiv2004

Representation functions of additive bases for abelian semigroups

Let X = S \oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \to N_0 cup infty be any map such that the set pi(f^{-1}(0)) is a finite subset of G. Then there exists a set B contained in X such that r_B(x) = f(x) for all x in X, where the representation function r_B(x) counts the number of sets {x',x''} contained in B such that x' \neq x'' and x'+x''=x. In particular, every function f from the integers Z into N_0 \cup infty such that f^{-1}(0) is finite is the representation function of an asymptotic basis for Z.

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arXiv2003

Every function is the representation function of an additive basis for the integers

Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,...,a_h are in A and a_1 \leq a_2 \leq ... \leq a_h. The function r_{A,h}: Z \to N_0 \cup \infty is the representation function of order h for A. The set A is called an asymptotic basis of order h if r_{A,h}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of exactly h not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z \to N_0 \cup \infty is any function such that f^{-1}(0) is finite, then there exists a set A of integers such that f(n) = r_{A,h}(n) for all n in Z. Moreover, the set A can be arbitrarily sparse in the sense that, if ϕ(x) \to \infty, then there exists a set A with f(n) = r_{A,h}(n) such that card{a in A : |a| \leq x} < ϕ(x) for all sufficiently large x.

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arXiv2003

Quantum integers and cyclotomy

A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with rational coefficients that satisfy this functional equation.

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arXiv2003

A functional equation arising from multiplication of quantum integers

For the quantum integer $[n]_q = 1+q+...+q^{n-1}$ there is a natural polynomial multiplication $*_q$ such that $[m]_q *_q [n]_q = [mn]_q$. This multiplication leads to the functional equation $f_{mn}(q) = f_m(q)f_n(q^m),$ defined on a given sequence $\mathcal(F)=\{f_n(q)\}_{n=1}^{\infty}$ of polynomials. This paper contains various results concerning the classification and construction of polynomial sequences that satisfy the functional equation, as well as a list of open problems that arise fromthe classification.

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arXiv2003

The inverse problem for representation functions of additive bases

Let A be a set of integers. For every integer n, let r_{A,2}(n) denote the number of representations of n in the form n = a_1 + a_2, where a_1 and a_2 are in A and a_1 \leq a_2. The function r_{A,2}: Z \to N_0 \cup {\infty} is the representation function of order 2 for A. The set A is called an asymptotic basis of order 2 if r_{A,2}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of two not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z \to N_0\cup {\infty} is any function such that f^{-1}(0) is finite, then there exists a set A of integers such that f(n) = r_{A,2}(n) for all n \in \Z. Moreover, the set A can be constructed so that card{a\in A : |a| \leq x} \gg x^{1/3}.

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arXiv2003

On the ubiquity of Sidon sets

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1,2,...,n} are B_2[g]-sets, in the sense that if B_2[g](k,n) denotes the number of B_2[g]-sets of cardinality k contained in the interval {1,2,...,n}, then lim_{n\to\infty} B_2[g](k,n)/\binom{n}{k} = 1 if k = o(n^{g/(2g+2)}).

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arXiv2003

Formal power series arising from multiplication of quantum integers

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a sequence {f_n(q)} of polynomials such that f_n(0)=1 for all n. It is proved that if {f_n(q)} is a solution of this functional equation, then the sequence {f_n(q)} converges to a formal power series F(q). Quantum mulitplication also leads to the functional equation f(q)F(q^m) = F(q), where f(q) is a fixed polynomial or formal power series with constant term f(0)=1, and F(q)=1+\sum_{k=1}^{\infty}b_kq^k is a formal power series. It is proved that this functional equation has a unique solution F(q) for every polynomial or formal power series f(q). If the degree of f(q)is at most m-1, then there is an explicit formula for the coefficients b_k of F(q) in terms of the coefficients of f(q) and the m-adic representation of k. The paper also contains a review of convergence properties of formal power series with coefficients in an arbitrary field or integeral domain.

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arXiv2003

Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture

Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let r_{A}(n,h) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,..., a_h are elements of A and a_1 \leq a_2 \leq ... \leq a_h. The infinite set A is called a basis of order h if r_{A}(n,h) \geq 1 for every nonnegative integer n. Erdos and Turan conjectured that limsup_{n\to\infty} r_A(n,2) = \infty for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problem, a special case of which is the Erdos-Turan conjecture. Konig's lemma on the existence of infinite paths in certain graphs is used to prove that this general problem is equivalent to a related problem about finite sets of nonnegative integers.

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arXiv2002

Polynomial growth of sumsets in abelian semigroups

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. 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 |hA| = p(h) for all sufficiently large h. Lattice point counting is also used to prove that sumsets of the form h_1A_1 + >... + h_rA_r have multivariate polynomial growth.

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arXiv2002

Additive number theory and the ring of quantum integers

Let $m$ and $n$ be positive integers. For the quantum integer $[n]_q = 1 + q + ... + q^{n-1}$ there is a natural polynomial addition such that $[m]_q \oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that $[m]_q \otimes_q [n]_q = [mn]_q$. These constructions lead to the construction of the ring of quantum integers and the field of quantum rational numbers. It is also shown that addition and multiplication of quantum integers are equivalent to elementary decompositions of intervals of integers in additive number theory.

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arXiv2002

Unique representation bases for the integers

Arbitrarily sparse sets A of integers are constructed with the property that every integer can be represented uniquely in the form n = a + a', where a and a' belong to the set A and a < a' or a = a'. Some related open problems are stated.

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arXiv2000

N-graphs, modular Sidon and sum-free sets, and partition identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.

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