Inverse Problems for Representation Functions in Additive Number Theory

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. Let SF(G) denotes the set of all sum-free subets of $G$ and $\sigma(G)$ denotes the number $ n^{-1}(\log_2 |SF(G)|) $. In this article we shall improve the error term in the asymptotic formula of $\sigma(G)$ which was obtained recently by Ben Green and Ruzsa. The methods used are a slight refinement of methods develop...

In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Zaremba's conjecture from number theory.

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...

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$ and $a\leq a'$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\geq2N-1$, then for any $\theta$ with $0<\theta<\frac{2\log2-\log3}{42\log 2-9\log3}$, the set of integers $n$ with $R_2(A,n)=\frac{n}{8}+O(n^{1-\theta})$ has density...

We establish analogues in the context of group actions or group representations of some classical problems and results in addtive combinatorics of groups. We also study the notion of left invariant submodular function defined on power sets which plays a central role in our proofs.

This is a survey of some of Erd\H os's work on bases in additive number theory.

We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics.

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{\frac{k-2}{k}-\epsilon})$ for some $\epsilon>0$, then $R_{\mathbb{N}\setminus A,k}(n)$ is eventually st...

A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant gamma=0.57721566.... This paper claims little originality, its...

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic f...