scholarly journals Restricted set addition in Abelian groups: results and conjectures

2005 ◽  
Vol 17 (1) ◽  
pp. 181-193 ◽  
Author(s):  
Vsevolod F. Lev
Keyword(s):  
Abelian Groups ◽  
Set Addition

scholarly journals Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture

10.37236/1482 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Vsevolod F. Lev
Keyword(s):  
General Type ◽  
Sharp Estimate ◽  
Abelian Groups ◽  
Polynomial Method ◽  
Set Addition

In 1980, Erdős and Heilbronn posed the problem of estimating (from below) the number of sums $a+b$ where $a\in A$ and $b\in B$ range over given sets $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ of residues modulo a prime $p$, so that $a\neq b$. A solution was given in 1994 by Dias da Silva and Hamidoune. In 1995, Alon, Nathanson and Ruzsa developed a polynomial method that allows one to handle restrictions of the type $f(a,b)\neq 0$, where $f$ is a polynomial in two variables over ${\Bbb Z}/p{\Bbb Z}$. In this paper we consider restricting conditions of general type and investigate groups, distinct from ${\Bbb Z}/p{\Bbb Z}$. In particular, for $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ and ${\cal R}\subseteq A\times B$ of given cardinalities we give a sharp estimate for the number of distinct sums $a+b$ with $(a,b)\notin\ {\cal R}$, and we obtain a partial generalization of this estimate for arbitrary Abelian groups.

THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

2008 ◽  
Vol 04 (06) ◽  
pp. 927-958 ◽  
Author(s):  
ÉRIC BALANDRAUD
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Additive Number Theory ◽  
Alternating Groups ◽  
Additive Number ◽  
New Interpretation ◽  
Set Addition

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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