# diophantus

Hello, this is beta version of diophantus. If you want to report about a mistake, please, write to hello@diophantus.org
All articles by
Hamidoune Yahya Ould | Hamidoune Y. O. | Hamidoune Yahya O.

Yahya Keyvan | Yahya W. A. | Yahya Mohamed | Yahya Sahba | Yahya Ali | Yahya Iskandar | Yahya Khairun | Yahya Khawaja Muhammad | Yahya Muhammad | Yahya Nurhartini Kamalia

# Search results

#### Large restricted sumsets in general abelian group

10 May 2013 math.NT arxiv.org/abs/1305.2431

Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^S B).

#### Subset sums in abelian groups

08 Dec 2011 math.CO math.GR math.NT arxiv.org/abs/1112.1929

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Z_p to the case of a general finite abelian group. To do so, we adapt Olson's method using a generalisation of Vosper's Theorem proved by Hamidoune and Plagne.

#### k-Sums in abelian groups

10 Oct 2011 math.CO math.GR math.NT arxiv.org/abs/1110.1961

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A|

= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.

#### Topology of Cayley Graphs Applied to Inverse Additive Problems

08 Nov 2010 math.CO math.NT arxiv.org/abs/1011.1797

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\le r \le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).

#### On dilates sums

23 May 2010 math.NT arxiv.org/abs/1005.4233

Let $A$ be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum $|A+k\cdot A|,$ where $k$ is a prime and $|A|$ is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let $k$ be an odd prime and assume that $|A|>8k^{k}.$ A corollary to our main result states that $|2\cdot A+k\cdot A|\ge (k+2)|A|-k^2-k+2.$ Notice that $|2\cdot P+k\cdot P|=(k+2)|P|-2k,$ if $P$ is an arithmetic progression.

#### On Minkowski product size: The Vosper's property

18 Apr 2010 math.CO arxiv.org/abs/1004.3010

A subset $S$ of a group $G$ is said to be a Vosper's subset if $|A\cup AS|\ge \min (|G|-1,|A|+|S|),$ for any subset $A$ of $G$ with $|A|\ge 2.$ In the present work, we describe Vosper's subsets. Assuming that $S$ is not a progression and that $|S^{-1} S|, |S S^{-1}| <2 |S|,|G'|-1,$ we show that there exist an element $a\in S,$ and a non-null subgroup $H$ of $G'$ such that either $S^{-1}HS =S^{-1}S \cup a^{-1}Ha$ or $SHS^{-1} =SS^{-1}\cup aHa^{-1},$ where $G'$ is the subgroup generated by $S^{-1}S.$

#### Extensions of the Moser-Scherck-Kemperman-Wehn Theorem

10 Feb 2009 math.CO math.NT arxiv.org/abs/0902.1680

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)={v}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$ (ii) If $F$ is cofinite, then $| \Gamma (F)\setminus F|\ge |\Gamma ^- (v)|-1.$ In particular, let $G$ be group, $B$ be a finite subset of $G$ and let $F$ be a finite or a cofinite subset of $G$ such that $F\cap B^{-1}={1}$. Then $| (FB)\setminus F|\ge |B|-1.$ The last result (for $F$ finite), is famous Moser-Scherck-Kemperman-Wehn Theorem. Its extension to cofinite subsets seems new. We give also few applications.

#### On Group bijections $\phi$ with $\phi(B)=A$ and $\forall a\in B, a\phi(a) \notin A$

13 Dec 2008 math.CO arxiv.org/abs/0812.2522

A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\phi : S \to T$ such that $x\phi(x)\notin T,$ for every $x\in S.$ The number of such pairings will be denoted by $\mu(S,T)$. Let $A$ and $B$ be finite subsets of a group $G$ with $1\notin B$ and $|A|=|B|.$ Also assume that the order of every element of $B$ is $\ge |B|$. Extending results due to Losonczy and Eliahou-Lecouvey, we show that $\mu(B,A)\neq 0.$ Moreover we show that $\mu(B,A)\ge \min {\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}},$ unless there is $a\in A$ such that $|Aa^{-1}\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular, either $\mu(B,B) \ge \min {\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}},$ or for some $a\in B,$ $Ba^{-1}$ is a progression.

#### A Structure Theory for Small Sum Subsets

19 Nov 2008 math.NT arxiv.org/abs/0811.3061

We develop a new method leading the structure of finite subsets S and T of an abelian group with $|S+T|\le |S|+|T|$. We show also how to recover the known results in this area in a relatively short space.

#### Hyper-atoms and the critical pair Theory

22 May 2008 math.NT arxiv.org/abs/0805.3522

We introduce the notion of a hyper-atom. One of the main results of this paper is the $\frac{2|G|}3$--Theorem: Let $S$ be a finite generating subset of an abelian group $G$ of order $\ge 2$. Let $T$ be a finite subset of $G$ such that $2\le |S|\le |T|$, $S+T$ is aperiodic, $0\in S\cap T$ and $$\frac{2|G|+2}3\ge |S+T|= |S|+|T|-1.$$ Let $H$ be a hyper-atom of $S$. Then $S$ and $T$ are $H$--quasi-periodic. Moreover $\phi(S)$ and $\phi(T)$ are arithmetic progressions with the same difference, where $\phi :G\mapsto G/H$ denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman's Structure Theorem.