ID: 0712.0408

Inverse Problems for Representation Functions in Additive Number Theory

December 3, 2007

View on ArXiv

Similar papers 2

Representation functions of additive bases for abelian semigroups

November 13, 2002

88% Match
Melvyn B. Nathanson
Number Theory
Combinatorics

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''} con...

Find SimilarView on arXiv

On The Determination of Sets By Their Subset Sums

January 11, 2023

88% Match
Andrea Ciprietti, Federico Glaudo
Number Theory
Combinatorics
Group Theory

Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be the multiset containing the $2^{|A|}$ sums of all subsets of $A$. We study the reconstruction problem ``Given $FS(A)$, is it possible to identify $A$?'', and we give a satisfactory answer for all abelian groups. We prove that, up to identifying multisets through a natural equivalence relation, the function $A \mapsto FS(A)$ is injective (and thus the reconstruction problem is solvable) if and only if ev...

Find SimilarView on arXiv

Weighted representation functions on $\mathbb{Z}_m$

August 21, 2012

88% Match
Quan-Hui Yang, Yong-Gao Chen
Number Theory
Combinatorics

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For any set $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(A,n)=\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)$ for all $n\in \mathbb{Z}_m$. We also pose several problems for further resear...

Find SimilarView on arXiv

Generalizations of some results about the regularity properties of an additive representation function

April 20, 2018

88% Match
Sándor Z. Kiss, Csaba Sándor
Number Theory

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os proved that if $\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty$ then $|\Delta_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $\Delta_{l}$ denot...

Find SimilarView on arXiv

On the Inverse Erdos-Heilbronn Problem for Restricted Set Addition in Finite Groups

October 24, 2012

87% Match
Suren M. Jayasuriya, Steven D. Reich, Jeffrey Paul Wheeler
Combinatorics

We provide a survey of results concerning both the direct and inverse problems to the Cauchy-Davenport theorem and Erdos-Heilbronn problem in Additive Combinatorics. We formulate an open conjecture concerning the inverse Erdos-Heilbronn problem in nonabelian groups. We extend an inverse to the Dias da Silva-Hamidoune Theorem to Z/nZ where n is composite, and we generalize this result into nonabelian groups.

Find SimilarView on arXiv

A generalization of sumsets modulo a prime

January 26, 2015

87% Match
Francesco Monopoli
Number Theory
Combinatorics

Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If $G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when $G=\mathbb{Z}/p\mathbb{Z}$ ...

Find SimilarView on arXiv

Inverse problems in Additive Number Theory and in Non-Abelian Group Theory

March 12, 2013

87% Match
G. A. Freiman, M. Herzog, P. Longobardi, ... , Stanchescu Y. V.
Number Theory
Combinatorics
Group Theory

The aim of this paper is threefold: a) Finding new direct and inverse results in the additive number theory concerning Minkowski sums of dilates. b) Finding a connection between the above results and some direct and inverse problems in the theory of Baumslag-Solitar (non-abelian) groups. c) Solving certain inverse problems in Baumslag-Solitar groups or monoids, assuming appropriate small doubling properties.

Find SimilarView on arXiv

Direct and Inverse Theorems on Signed Sumsets of Integers

October 5, 2018

87% Match
Jagannath Bhanja, Ram Krishna Pandey
Number Theory

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots, \lambda_{k-1}) \in \mathbb{Z}^{k},~ \Sigma_{i=0}^{k-1}|\lambda_{i}|=h \},\] be the {\it signed sumset} of $A$. The {\it direct problem} for the signed sumset $h_{\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\underline{+}}A|$ in terms...

Find SimilarView on arXiv

A Walk Through Some Newer Parts of Additive Combinatorics

November 3, 2022

87% Match
Bela Bajnok
Number Theory

In this survey paper we discuss some recent results and related open questions in additive combinatorics, in particular, questions about sumsets in finite abelian groups.

Find SimilarView on arXiv

Additive combinatorics with a view towards computer science and cryptography: An exposition

August 18, 2011

87% Match
Khodakhast Bibak
Combinatorics
Cryptography and Security
Number Theory

Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define - perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremend...

Find SimilarView on arXiv