# 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.

32 questions
Filter by
Sorted by
Tagged with
102 views

• 21.5k
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 ...
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 ...
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 $d_i−d_j \pmod{q}$ ...
• 3,331
1 vote
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 ...
• 3,331
33 views

### 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 ...
• 3,331
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 ...
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 ...
• 111
1 vote
88 views

### 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 ...
• 29
87 views

• 2,449
442 views

• 3,620
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: $$v=|G|=\dfrac{q^{d+1}-1}{q-1}.$$ ...
• 3,620
1 vote
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 ...
• 3,620
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 ...
• 2,449
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 $A$. ...
• 8,596
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 ...
• 411
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 $|S - S| = 2 |S| - 1$,...
• 4,888
1 vote
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 ...
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 $S-S$ ...
• 12.8k
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 ...
• 738
1 vote
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 $d_1d_2^{-1}$ ...
• 2,012
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 ...
• 85
1 vote