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Questions tagged [difference-sets]

For questions about difference sets of groups, their developments to symmetric designs, their existence and non-existence, multipliers, and other properties.

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Why is the difference of consecutive primes from Fibonacci sequence divisible by $4$?

The primes represented in the Fibonacci sequence are written in the form $6n + 1$ and $6n -1$, respectively. $$5=6\times1-1$$ $$13=6\times2+1$$ $$89=6\times15-1$$ $$233=6\times39-1$$ $$1597=6\times266+...
Polona Čuk Kozoderc's user avatar
0 votes
1 answer
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If set of divisors of elements of a difference set $\Delta S = S - S \subset R$ a commutative ring is the whole ring (!) then $\Delta S$ is an ideal

Let $R$ be an infinite commutative ring. Let $S \subset R$ be such that $\Delta S = \{ s - s' : s,s' \in S\}$ has the properties: $$ \forall a \in R\setminus 0,\ a \textbf{ divides } g \textbf{ for ...
HighAsAKiteOnMath's user avatar
6 votes
1 answer
246 views

When can $L$ sets of the form $\{a,b,a+b\}$ partition $\{1,2,\dots, 3L\}$?

Now also posted to MathOverflow. Consider a set of the form $\{a,b,a+b\}$ where $a$ and $b$ are positive integers with $b > a$. I will refer to such a set as a triplet. Consider now the problem of ...
Mohannad Shehadeh's user avatar
2 votes
1 answer
123 views

Set of integers with unique differences

I have looked at the interesting Conway-Guy Sequence which possesses a neat property of having unique subset sums. But I would like to find an integer set (which is optimally compact, ie has the ...
Parsa IQT's user avatar
0 votes
0 answers
64 views

Existence of a pair of elements in a Perfect Difference Set

Perfect Difference Sets (PDS) of order $m+1$ are a set of residues $\{d_1,d_2,\cdots,d_{m+1}\} \pmod{q}$ such that every non-zero residue modulo $q$ can be uniquely represented by $d_i−d_j \pmod{q}$ ...
vvg's user avatar
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1 vote
0 answers
32 views

A congruence constraint for constructing Perfect Difference Sets

I have been reading on Perfect Difference Sets and it has been fascinating. I have an open MSE question here seeking references for algorithms to construct PD sets. This is a different question ...
vvg's user avatar
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Reference request: Algorithms for constructing Perfect Difference Sets

I am looking for algorithms to construct Perfect Difference Sets. Perfect Difference Sets (PDS) of order $m+1$ are a set of residues $\{d_1, d_2, \cdots, d_{m+1}\} \pmod {q}$ such that every non-zero ...
vvg's user avatar
  • 3,331
0 votes
2 answers
54 views

Characterization of density through the interior

I'm trying to prove an elemental exercise of general topology but I don't know how: Let $C = B \setminus A$, then $A$ is dense in $B$ if and only if $\operatorname{int}(C) = \varnothing$. Any hints or ...
Alex Aldrin's user avatar
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0 answers
51 views

What is that called in set theory that can be used in programming to make set A equal to set B by adding to and subtracting from B the differences?

What is that called in set theory that can be used in programming to make set B equal to set A by (1) adding to set B what set A has but set B does not have, and (2) removing from set B what set A ...
daniel's user avatar
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1 vote
0 answers
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How to convert sets of ODE to difference equations

I have successful developed a continuous mathematical model for my physical system. The model comprises of sets of 7 nonlinear ODE. My objective is to transform the continuous-time ODE into a set of ...
Tee's user avatar
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2 votes
1 answer
87 views

Largest set $B$ such that $|A\cap (B-B)|=p$

In a preprint I was reading the following was claimed without proof: Let $A$ be a subset of $[n]:=\{1,2,\dots n\}$ where $|A|<\frac{n}{k}$ for some integer $k$. Then there exists a set $B\subset [...
Zach Hunter's user avatar
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2 votes
2 answers
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How do you find a $(21,5,1)$-difference set in $(\mathbb{Z}_{21}, +)$?

How do you find a $(21,5,1)$-difference set in $(\mathbb{Z}_{21}, +)$? I already know the answer which is $\{0,1,6,8,18\}$. But How do you get that? Obviously, if you subtract each elements by ...
Ka Em's user avatar
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1 vote
1 answer
41 views

Subsets of real numbers satisfying the two conditions

Is there any uncountable subset $B$ of real numbers such that: (1) $(B-B)\cap (-1,1)=\{ 0\}$, (2) $(-1,1)+B=\mathbb{R}$? Also, what is the answer if $(-1,1)$ is replaced by $[-1,1)$, $(-1,1]$ or $[-...
M.H.Hooshmand's user avatar
-5 votes
1 answer
442 views

Writing down the following sets [closed]

Given that $\Bbb R$ denotes the set of all real numbers, $\Bbb Z$ the set of all intergers, and $\Bbb Z^-$ the set of all negative integers, describe each of the following sets. a. $\{x\in\Bbb R \mid ...
Jlynz's user avatar
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1 vote
0 answers
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A problem of sum set and difference set

Given a set of $N$ distinct positive integers $ {A}=\left\{a_1,a_2,\cdots,a_N\right\}$. Denote the sum set of elements in $A$ as $B=\left\{a_m+a_n|a_m,a_n\in A \right\}$. Then, denote the difference ...
abao5887's user avatar
2 votes
1 answer
174 views

Efficient calculation of difference sets from finite fields

A while ago I wrote a program to generate, amongst other things, difference sets from finite fields. Generating these sets is rather slow. Is there some theorem or construction I could use to speed it ...
del42z's user avatar
  • 85
3 votes
1 answer
140 views

Proof that the symmetric design isomorphic to $PG(m,q)$ has an automorphism acting regularly on points (Singer's Theorem)

I have annoyed various faculty members enough with this, so I will turn to you to check that my proof of Singer's Theorem is satisfactory. I have been politely told before on Math.SE that I give way ...
The Count's user avatar
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1 vote
1 answer
112 views

Is there a canonical form for difference sets?

A subset $D$ of group $G$ is called a $(v,k,\lambda)$-difference set if each element of $G$ (except identity) can be expressed as difference of $D$ elements ($d_1-d_2$) in exactly $\lambda$ ways. $v$ ...
Džuris's user avatar
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2 votes
1 answer
115 views

Check my logic: Does a generalized dihedral extension necessarily contain the extension element? (I say yes.)

I want to make sure I am correct about something I read in Moore and Pollatsek's "Difference Sets": Suppose we have an abelian group $H$. If $\exists g\notin H$ with $g^2=1$ and $ghg^{-1}=h^{-1}\;...
The Count's user avatar
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2 votes
1 answer
79 views

Groups of a specific order from a difference set.

So I am reading some surveys about Design Theory, and I am in over my head to a good extent. The text mentions a cyclic difference set in a group of order: $$v=|G|=\dfrac{q^{d+1}-1}{q-1}.$$ ...
The Count's user avatar
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1 vote
1 answer
125 views

Checking my understanding of Multipliers for Difference Sets

In "Contemporary Design Theory: A Collection of Surveys," pg. 245 begins the section on multipliers of difference sets. I had previously understood a multiplier $\alpha$ of a difference set $D$ in a ...
The Count's user avatar
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3 votes
1 answer
322 views

About difference set of Fibonacci numbers

Let $\mathcal{F}$ be the set of all Fibonacci numbers (defined by $ F_n=F_{n-1}+F_{n-2}$ with $F_1=F_2=1$), and put $D:=\mathcal{F}-\mathcal{F}$. (a) Is it true that if $B$ is a finite subsetset of ...
M.H.Hooshmand's user avatar
18 votes
1 answer
1k views

An old question about sumsets and difference sets

Let $A$ be a finite set. Define the symbols $+$ and $-$ as follows: $$A+A=\{a+b:a,b\in A\};$$ $$A-A=\{a-b:a,b\in A\}.$$ Prove or disprove $|A+A|\leq|A-A|$, where $|A|$ denotes the cardinality of $A$. ...
Yuxiao Xie's user avatar
  • 8,596
0 votes
1 answer
130 views

Constructing $\lambda$-difference sets

Given a set say $A=\{0,1,4,16,r\}$ which is a subset of $\mathbb{Z}_{21}$. How do I find $r$, such that $A$ is a $\lambda$-difference set for some $\lambda$? Is there some methodical way to solve ...
vounoo's user avatar
  • 411
8 votes
1 answer
322 views

Are there any "nontrivial" sets with small difference sets?

I'm trying to find finite sets $S$ of natural numbers with "small" difference sets. One option is just taking an arithmetic progression $S = \{0, , \ldots, n-1\}$. Then $|S - S| = 2 |S| - 1$,...
Julien Clancy's user avatar
1 vote
0 answers
37 views

Finding an imperfect finite difference set for large N

I want to find a set of integers $N$ for which there always exists a pair of numbers $(a, b)$ both $\in N$ such that $a-b = x$ for all $0<x<2^{32}$. Obviously one possible set N is all the ...
user136176's user avatar
2 votes
1 answer
289 views

Difference sets avoiding quadratic residues

I have a homework question that is stumping me, and I am looking for an entry point. It goes like this: Suppose $p$ is prime. Prove that the largest set $S\subseteq\{0,1,\dots, p-1\}$ such that $S-S$ ...
Eric Stucky's user avatar
  • 12.8k
2 votes
1 answer
190 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
KcH's user avatar
  • 738
1 vote
2 answers
938 views

Show that the complement of a difference set is a difference set

In combinatorics, a $(v,k,\lambda)$ difference set is a subset $D$ of cardinality $k$ of a group $G$ of order $v$ such that every nonidentity element of $G$ can be expressed as a product $d_1d_2^{-1}$ ...
Alen's user avatar
  • 2,012
3 votes
0 answers
375 views

The relation between perfect difference sets and finite projective planes

Given a (finite) perfect difference set, it is easy to create a finite projective plane. I'm wondering: Given a finite projective plane, does there necessarily exist a corresponding perfect ...
del42z's user avatar
  • 85
1 vote
2 answers
196 views

Constructions of small set with big difference set

Does anyone know any constructions of a small set with a big difference set? Mathematically speaking: Let $A\subseteq \mathbb{Z}$, such that $A-A=\mathbb{Z}_n$. Please give a sequence $(A_n)_{n\in \...
Terry Zhou's user avatar
1 vote
1 answer
502 views

Combinatorial design difference set proof: adding 0 to Paley difference set

Suppose $D$ is the $(4t-1,2t-1,t-1)$-difference set obtained from the following theorem: Suppose $v$ is a prime power congruent to $3\bmod 4$. Let $v=4t-1$. Then there is a $(4t-1,2t-1,t-1)$-...
Sarah Martin's user avatar