scholarly journals The Sharp Threshold for Maximum-Size Sum-Free Subsets in Even-Order Abelian Groups

2015 ◽  
Vol 24 (4) ◽  
pp. 609-640 ◽  
Author(s):  
NEAL BUSHAW ◽  
MAURÍCIO COLLARES NETO ◽  
ROBERT MORRIS ◽  
PAUL SMITH
Keyword(s):  
Recent Work ◽  
Abelian Groups ◽  
Maximum Size ◽  
Constant Factor ◽  
Sharp Threshold ◽  
Even Order ◽  
Free Subset ◽  
Free Sets

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the caseG= ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

scholarly journals ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

2009 ◽  
Vol 05 (06) ◽  
pp. 953-971 ◽  
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Finite Abelian Groups ◽  
Free Subset ◽  
Free Set ◽  
Free Sets

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group ℤn was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of λ3,1(ℤn). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when k - l and the order of G are relatively prime; we extend their results to see what happens without this assumption.

THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

2018 ◽  
Vol 99 (2) ◽  
pp. 184-194
Author(s):  
BÉLA BAJNOK ◽  
RYAN MATZKE
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Critical Pair ◽  
Cyclic Groups ◽  
Free Subset ◽  
Free Sets

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

scholarly journals On solving systems of random linear disequations

2008 ◽  
Vol 8 (6&7) ◽  
pp. 579-594
Author(s):  
G. Ivanyos
Keyword(s):  
Abelian Groups ◽  
Main Idea ◽  
Constant Factor ◽  
Linear Functions ◽  
Important Special Case ◽  
Hidden Subgroup Problem ◽  
Left Hand ◽  
Shift Problem ◽  
Group A ◽  
Subgroup Problem

An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

scholarly journals Representations of constant socle rank for the Kronecker algebra

2020 ◽  
Vol 32 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Daniel Bissinger
Keyword(s):  
Recent Work ◽  
Abelian Groups ◽  
Wild Type ◽  
Kronecker Algebra ◽  
Radical Property

AbstractInspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups {E_{r}} of rank r over a field of characteristic {p>0}, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra {\mathcal{K}_{r}=k\Gamma_{r}} with {r\geq 2} arrows and {1\leq d\leq r-1}. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in {\operatorname{mod}\mathcal{K}_{r}} are of wild type. Then we use a natural functor {\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}} to transfer our results to {\operatorname{mod}kE_{r}}.

Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups

2019 ◽  
Vol 29 (6) ◽  
pp. 401-407
Author(s):  
Aleksandr A. Sapozhenko ◽  
Vahe G. Sargsyan
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Groups ◽  
Free Sets

Abstract A family (A1, …, Ak) of subsets of a group G is called k-solution-free family if the equation x1 + … + xk = 0 has no solution in (A1, …, Ak) such that x1 ∈ A1, …, xk ∈ Ak. We find the asymptotic behavior for the logarithm of the number of k-solution-free families in Abelian groups.

scholarly journals Maximal sum-free sets in finite abelian groups, V

1975 ◽  
Vol 13 (3) ◽  
pp. 337-342 ◽  
Author(s):  
H.P. Yap
Keyword(s):  
Prime Divisor ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Free Set ◽  
Free Sets

Let λ(G) be the cardinality of a maximal sum-free set in a group G. Diananda and Yap conjectured that if G is abelian and if every prime divisor of |G| is congruent to 1 modulo 3, then λ(G) = |G|(n−1)/3n where n is the exponent of G. This conjecture has been proved to be true for elementary abelian p−groups by Rhemtulla and Street ana for groups by Yap. We now prove this conjecture for groups G = Zpq ⊕ Zp where p and q are distinct primes.

scholarly journals A new critical pair theorem applied to sum-free sets in Abelian groups

2004 ◽  
Vol 79 (1) ◽  
pp. 183-207 ◽  
Author(s):  
Yahya ould Hamidoune ◽  
Alain Plagne
Keyword(s):  
Abelian Groups ◽  
Critical Pair ◽  
Free Sets

scholarly journals Maximal sum-free sets in abelian groups of order divisible by three: Corrigendum

1972 ◽  
Vol 7 (2) ◽  
pp. 317-318 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Arithmetic Progression ◽  
Abelian Groups ◽  
Factor Group ◽  
Image Position ◽  
Free Set ◽  
Free Sets

The last step of the proof in [2] was omitted. To complete the argument, we proceed in the following way. We had shown that H = H(S) = H(S+S) = H(S-S), that |S-S| = 2|S| - |H| and hence that in the factor group G* = G/H of order 3m, the maximal sum-free set S* = S/H and its set of differences S* - S* are aperiodic, withso thatBy (1) and Theorem 2.1 of [1], S* - S* is either quasiperiodic or in arithmetic progression.

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