# Questions tagged [sidon-sets]

The sidon-sets tag has no usage guidance.

18
questions

1
vote

0
answers

304
views

### On fifth powers forming a Sidon set

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...

6
votes

3
answers

337
views

### Is this set of numbers independent/Sidon?

Given a subset $E\subset \mathbb N\setminus \{0\}$ of integers, we say that $E$ is independent if for any choice of $k\in \mathbb N$, $\epsilon_i\in\{-1,0,1\}$ for $i=1,\dots,k$, if $\Sigma_{i=1}^k\...

4
votes

1
answer

344
views

### Difference set of difference set

I am a hobby computer scientist and searching for an algorithm to construct a set of n numbers (integers) with certain properties.
Property 1 / Step 1
All pairwise differences of the elements should ...

4
votes

1
answer

268
views

### Unique representation and sumsets

Let $A$ be a finite, nonempty subset of an abelian group, and let $2A:=\{a+b\colon a,b\in A\}$ and $A-A:=\{a-b\colon a,b\in A\}$ denote the sumset and the difference set of $A$, respectively.
If ...

10
votes

1
answer

537
views

### Sidon sets of $\mathbb{Z}/p\mathbb{Z}$

A set $S \subseteq \mathbb{Z}/p\mathbb{Z}$ is called a Sidon set if given $a, b, c, d \in S$ and $a+ b = c+ d$, then $\{a, b\} = \{c,d\}$. I was interested in knowing about the largest possible Sidon ...

19
votes

4
answers

862
views

### Size of sets with complete double

Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My ...

5
votes

1
answer

161
views

### Eccentricity in the number of representations for sets too large to be Sidon sets

Let $A=\{a_1<a_2<a_3<\dots< a_k\}\subset\{1,2,\dots,N\}$ be a set of integers. Let $r_A(n)=\#\{(a_i,a_j):a_i+a_j=n\}$ be the number of representations of $n$ as a sum of two elements from $...

2
votes

0
answers

140
views

### Primality radii and Sidon sets

I learned tonight what a Sidon set is, in a book about Erdős. This notion inspires me the following question :
For $n$ a large enough composite integer, say $r>0$ is a primality radius of $n$ if ...

0
votes

0
answers

100
views

### Number of $b$-separated Sidon sets with pairwise difference set intersection bounds

Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...

4
votes

2
answers

382
views

### An extremal combinatorics problem

Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ what is the largest $m$ we have such that in the interval $(p^\alpha,p-p^\alpha)$ there are $m$ integers $a_1<\dots<a_m$ ...

1
vote

0
answers

49
views

### Has anyone studied Golomb rulers having a spectrum with a minimal $L^2$ norm?

A Golomb ruler can be described as a set of marks on a line having integer positions, such that no two pairs of marks are separated by the same distance. Call the spectrum of a ruler the (multi-)set ...

3
votes

1
answer

349
views

### Lower bound for k-fold Sidon Sets

k-fold Sidon set is defined in http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r25/pdf (page #4, paragraph 4)
Does anyone know what the best known lower bound construction is for the ...

3
votes

3
answers

494
views

### How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.
Example with $|A|=3$: Out of the set $A :...

4
votes

1
answer

155
views

### $B_k[1]$ sets with smallest possible $m = \max B_k[1]$ for given $k$ and $n = \lvert B_k[1]\rvert$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds
$$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}.$$
Thus if you know the sum of two elements, you know which elements ...

5
votes

3
answers

989
views

### Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums

I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include ...

6
votes

1
answer

366
views

### Recovering Sidon sets from difference sets, part 2.

This is inspired by a recent question. A set $A \subset \mathbb{Z}/n\mathbb{Z}$ with $|A|=m$ is a Sidon set if all the pairwise sums of distinct elements are unequal: $A+A=\{a+a' \mid a,a' \in A, a \...

2
votes

1
answer

569
views

### Recovering Sidon sets from difference sets

How can I recover a Sidon set $A\subseteq \mathbb{Z}/n\mathbb{Z}$ from the set $A-A\subseteq \mathbb{Z}/n\mathbb{Z}$?
Is it even unique? (up to translation and reflection)
($A-A$ stands for the set ...

3
votes

1
answer

426
views

### Optimize / simple Set Covering Problem

Let $k,m\in\mathbb{N}$ be given. Let $M:=\{0,... , m-1\}$. How to find a subset $T\subset M$, $|T|=k$ such that $|T+T|$ is maximal, where $T+T=\{ (a+b)\mathbin\%m \mid a\in T,b\in T \}$ (“%” means ...