Another proof of a theorem on difference sets

1967 ◽  
Vol 63 (3) ◽  
pp. 595-596
Author(s):  
D. L. Yates
Keyword(s):  
Cyclic Group ◽  
Elementary Proof ◽  
Existence Theorems ◽  
Difference Sets ◽  
Difference Set ◽  
En Bloc ◽  
Elementary Matrix

Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

The Multiplier Theorem for Difference Sets

1964 ◽  
Vol 16 ◽  
pp. 386-388 ◽  
Author(s):  
Richard J. Turyn
Keyword(s):  
Cyclic Group ◽  
Prime Divisor ◽  
Difference Sets ◽  
Difference Set ◽  
Multiplier Theorem

In a recent paper (5) Newman proved the following theorem: if D is a difference set in a cyclic group G and n = q is prime, then q is a multiplier of D. If n = 2q and (v, 7) = 1, then q is a multiplier of D. The purpose of this note is to point out that a stronger statement than the first part was proved in (1), to remove the restriction (v, 7) = 1 in the second part, and to give again and make some comments about the proof of the theorem which asserts that a prime divisor of n is a multiplier of D if prime to v.

Divisible Semiplanes, Arcs, and Relative Difference Sets

1987 ◽  
Vol 39 (4) ◽  
pp. 1001-1024 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Projective Plane ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
List Type ◽  
Relative Difference Set ◽  
Large Collection ◽  
Baer Subplane ◽  
Relative Difference Sets ◽  
Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

scholarly journals An Elementary Proof of Jin's Theorem with a Bound

10.37236/3846 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
Difference Sets ◽  
Explicit Bound ◽  
Finite Set

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If $A$ and $B$ are sets of integers having positive upper Banach densities $a$ and $b$ respectively, then there exists a finite set $F$ of cardinality at most $1/ab$ such that $(A-B)+F$ covers arbitrarily long intervals.

The Inverse Multiplier for Abelian Group Difference Sets

1964 ◽  
Vol 16 ◽  
pp. 787-796 ◽  
Author(s):  
E. C. Johnsen
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Collineation Group ◽  
Difference Sets ◽  
Difference Set ◽  
The One ◽  
A Finite Group

In(1)Bruck introduced the notion of a difference set in a finite group. LetGbe a finite group ofvelements and let D = {di},i= 1, . . . ,kbe ak-subset ofGsuch that in the set of differences {di-1dj} each element ≠ 1 inGappears exactly λ times, where 0 < λ <k<v— 1. When this occurs we say that (G,D) is av,k,λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that av,k, λ group difference set is equivalent to a transitivev,k,λconfiguration, i.e., av,k,λconfiguration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parametersv,kandλ, then, we have the relation shown by Ryser(5)

scholarly journals Nonabelian Groups with $(96,20,4)$ Difference Sets

10.37236/927 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Omar A. AbuGhneim ◽  
Ken W. Smith
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Existence Problem ◽  
Normal Subgroups ◽  
Nonabelian Groups

We resolve the existence problem of $(96,20,4)$ difference sets in 211 of 231 groups of order $96$. If $G$ is a group of order $96$ with normal subgroups of orders $3$ and $4$ then by first computing $32$- and $24$-factor images of a hypothetical $(96,20,4)$ difference set in $G$ we are able to either construct a difference set or show a difference set does not exist. Of the 231 groups of order 96, 90 groups admit $(96,20,4)$ difference sets and $121$ do not. The ninety groups with difference sets provide many genuinely nonabelian difference sets. Seven of these groups have exponent 24. These difference sets provide at least $37$ nonisomorphic symmetric $(96,20,4)$ designs.

Some New Difference Sets

1962 ◽  
Vol 14 ◽  
pp. 614-625 ◽  
Author(s):  
Basil Gordon ◽  
W. H. Mills ◽  
L. R. Welch
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Image Position

A difference set is a set D = {d1, d2, … , dk] of k distinct residues modulo v such that each non-zero residue occurs the same number of times among the k(k — 1) differences di — dj, i ≠ j. If λ is the number of times each difference occurs, then(1)When we wish to emphasize the particular values of v, k, and λ involved we will call such a set a (v, k, λ) difference set. Another (v, k, λ) difference set E = {e1, e2, … ek} is said to be equivalent to the original one if there exist a and t such that (t, v) = 1 and E = {a + td1, … , a + tdk}. If t = 1 we will call the set E a slide of the set D. If D = E, then t is called a multiplier of D.

Multipliers of Difference Sets

1963 ◽  
Vol 15 ◽  
pp. 121-124 ◽  
Author(s):  
Morris Newman
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Residue Modulo ◽  
Image Position

Let λ, K, v be integers such that 0 < λ < k < v. Then the set of integersis a difference set with parameters v, k, λ if each non-zero residue modulo v occurs precisely λ times and zero occurs precisely k times among the k2 numbers

scholarly journals Difference sets and frequently hypercyclic weighted shifts

2013 ◽  
Vol 35 (3) ◽  
pp. 691-709 ◽  
Author(s):  
FRÉDÉRIC BAYART ◽  
IMRE Z. RUZSA
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Hypercyclic Operators ◽  
Weighted Shifts ◽  
Frequently Hypercyclic Operators ◽  
Upper Density ◽  
The Difference ◽  
Positive Integers

AbstractWe solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on${\ell }^{p} ( \mathbb{Z} )$,$p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is$ \mathcal{U} $-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on${c}_{0} $. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

A Family of Difference Sets

1958 ◽  
Vol 10 ◽  
pp. 73-77 ◽  
Author(s):  
R. G. Stanton ◽  
D. A. Sprott
Keyword(s):  
Difference Sets ◽  
Difference Set ◽  
Ordered Pairs

A difference set (,D) is defined in (2) as a subset D of k elements in a group of order υ with the following properties : (1) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (d 1 d 2) of elements of D such that x = 1 d 2; (2) if x ∊ , x ≠ 1, there are exactly λ distinct ordered pairs (3 d 4) of elements of D such that x = 3 d 4−1.

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