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

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

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

Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory

The set A = {a_n} of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If a_n ~ alpha n^h for some real number alpha > 0, then alpha is called an additive eigenvalue of order h. The additive spectrum of order h is the set N(h) consisting of all additive eigenvalues of order h. It is proved that there is a positive number eta_h <= 1/h! such that N(h) = (0, eta_h) or N(h) = (0, eta_h]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers B = {b_n} such that b_n ~ 2^n and B contains a subsequence {b_{n_k}} such that b_{n_k} ~ 3^k.Comment: 12 pages; minor revision

Desperately seeking mathematical truth

This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.Comment: This article has been published on the Opinion page of the Notices of the American Mathematical Society, August, 200

Representation functions of bases for binary linear forms

Let F(x_1,...,x_m) = u_1 x_1 + ... + u_m x_m be a linear form with nonzero, relatively prime integer coefficients u_1,..., u_m. For any set A of integers, let F(A) = {F(a_1,...,a_m) : a_i in A for i=1,...,m}. The representation function associated with the form F is R_{A,F}(n) = card {(a_1,...,a_m) in A^m: F(a_1,..., a_m) = n}. The set A is a basis with respect to F for almost all integers the set Z\F(A) has asymptotic density zero. Equivalently, the representation function of an asymptotic basis is a function f:Z -> N_0 U {\infty} such that f^{-1}(0) has density zero. Given such a function, the inverse problem for bases is to construct a set A whose representation function is f. In this paper the inverse problem is solved for binary linear forms.Comment: Improved version with some typos corrected; 8 page