Growth of sumsets in abelian semigroups

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define 0A = {0}. If A_1,...,A_r, and B are finite sumsets of A and h_1,...,h_r are nonnegative integers, the sumset h_1A + ... + h_rA_r + B is the set of all elements of S that can be represented in the form u_1 + ... + u_r + b, where u_i \in h_iA_i and b \in B. The growth function of this sumset is \gamma(h_1,...,h_r) = |h_1A + ... + h_rA_r + B|. Applying the Hilbert function for graded modules over graded algebras, where the grading is over the semigroup of r-tuples of nonnegative integers, we prove that there is a polynomial p(t_1,...,t_r) such that \gamma(h_1,...,h_r) = p(t_1,...,t_r) if min(h_1,...,h_r) is sufficienlty large.Comment: 5 pages. To appear in Semigroup Foru

Partitions with parts in a finite set

Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).Comment: 5 pages. To appear in the Proceedings of the American Mathematical Societ

Sumsets contained in sets of upper Banach density 1

Every set $A$ of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that $B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I} B_i \subseteq A$ for every nonempty finite set $I$ of positive integers.Comment: 7 pages; one additional theore

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 \phi(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} < \phi(x) for all sufficiently large x.Comment: 15 pages. LaTex file. Corrected and revised manuscrip

Heights on the finite projective line

Define the height function h(a) = min{k+(ka\mod p): k=1,2,...,p-1} for a = 0,1,...,p-1. It is proved that the height has peaks at p, (p+1)/2, and (p+c)/3, that these peaks occur at a= [p/3], (p-3)/2, (p-1)/2, [2p/3], p-3,p-2, and p-1, and that h(a) \leq p/3 for all other values of a.Comment: 10 pages, 1 figur

Every finite set of integers is an asymptotic approximate group

A set $A$ is an $(r,\ell)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq X+A$. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r \geq 2$.Comment: 7 pages; minor corrections and improvement

Comparison estimates for linear forms in additive number theory

Let $R$ be a commutative ring $R$ with $1_R$ and with group of units $R^{\times}$. Let $\Phi = \Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$ be an $h$-ary linear form with nonzero coefficients $\varphi_1,\ldots, \varphi_h \in R$. Let $M$ be an $R$-module. For every subset $A$ of $M$, the image of $A$ under $\Phi$ is $$\Phi(A) = \{ \Phi(a_1,\ldots, a_h) : (a_1,\ldots, a_h) \in A^h \}.$$ For every subset $I$ of $\{1,2,\ldots, h\}$, there is the subset sum $s_I = \sum_{i\in I} \varphi_i.$ Let $\mathcal{S} (\Phi) = \{s_I: \emptyset \neq I \subseteq \{1,2,\ldots, h\} \}.$ Theorem. Let $\Upsilon(t_1,\ldots, t_g) = \sum_{i=1}^g \upsilon_it_i$ and $\Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$ be linear forms with nonzero coefficients in the ring $R$. If $\{0, 1\} \subseteq \mathcal{S} (\Upsilon)$ and $\mathcal{S} (\Phi) \subseteq R^{\times}$, then for every $\varepsilon > 0$ and $c > 1$ there exist a finite $R$-module $M$ with $|M| > c$ and a subset $A$ of $M$ such that $\Upsilon(A \cup \{0\}) = M$ and $|\Phi(A)| < \varepsilon |M|$.Comment: 20 pages. Minor revision

Sidon sets and perturbations

Let $h$ be a positive integer. An $h$-Sidon set in an additive abelian group $G$ is a subset $A = \{a_i:i \in I \}$ of $G$ such that, if $a_{i_j} \in A$ for $j =1,\ldots, 2h$ and $a_{i_1} +\cdots + a_{i_h} = a_{i_{h+1}} + \cdots + a_{i_{2h}}$, then there is a permutation $\sigma$ of the set $\{1,\ldots, h\}$ such that $a_{i_{h+j}} = a_{i_{\sigma(j)}}$ for all $j \in \{1,\ldots, h\}$. It is proved that almost every finite set of real numbers is an $h$-Sidon set. Let $\varepsilon = (\varepsilon_i)_{i \in I}$, where $\varepsilon_i > 0$ for all $i \in I$. Let $\mathbf{F}$ be a field with a nontrivial absolute value. The set $B = \{b_i :i \in I \}$ in $\mathbf{F}$ is an $\varepsilon$-perturbation of the set $A = \{a_i :i \in I \}$ in $\mathbf{F}$ if $|b_i-a_i| < \varepsilon_i$ for all $i \in I$. It is proved that, for every $\varepsilon$, every countably infinite set $A$ has an $\varepsilon$-perturbation $B$ that is an $h$-Sidon set.Comment: 7 page

Matrix scaling limits in finitely many iterations

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter family of row but not column stochastic $n\times n$ positive matrices is constructed that become doubly stochastic after exactly one column scaling.Comment: 6 page

Nets in groups, minimum length gg-adic representations, and minimal additive complements

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.Comment: 16 page