All Questions
37
questions
2
votes
2
answers
60
views
Regarding scaling in sumsets
Let $A$ be a finite subset of $\mathbb{N}$. We denote the set $\{a_1 +a_2: a_1, a_2\in A\}$ as $2A$. We call the quantity $\sigma[A]:= |2A|/|A|$ as the doubling constant of $A$, and this constant can ...
- 971
2
votes
0
answers
111
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 9,534
1
vote
1
answer
66
views
Sum-free sequence but multiset
The question is:
Show that if $S$ is a set of natural numbers such that no number in S can be expressed as a sum of other (not necessarily distinct) numbers in S, then $\sum_{ s \in S} \frac{1}{s} \...
- 506
0
votes
0
answers
38
views
Schnirelmann density and bases of finite order
Let $\mathcal{A}$ be an additive set. We know that if the Schnirelmann density $\sigma_{\mathcal{A}}$ is positive then it is a basis of finite order. But it it not a necessary condition. My question ...
user1232857
1
vote
1
answer
88
views
If $A \subseteq \mathbb{Z}/p\mathbb{Z}$, and $|A| > \frac{p}{3}$, then are there any nontrivial lower bounds for $|AA|$?
If $A \subseteq \mathbb{Z}/p\mathbb{Z}$, and $|A| > \frac{p}{3}$, then are there any nontrivial lower bounds for $|AA|$?
Where $AA=\{a_{1} \cdot a_2:a_1,a_2 \in A\}$, and $p$ is prime.
Writing out ...
- 196
1
vote
0
answers
23
views
When is the Frobenius number of a numerical semigroup larger than the maximum of the minimal generating set
Let $S$ be a numerical semigroup (https://en.m.wikipedia.org/wiki/Numerical_semigroup). Let $A$ be the minimal generating set for $S$. As standard, let $e(S)$, $m(S)$ and $F(S)$ stand respectively ...
- 65
0
votes
1
answer
54
views
On writing every integer from $(a-1)(b-1)$ onwards as a sum of two non-zero integers from the semigroup generated by $a,b$
Let $\mathbb N$ be the semigroup (even a monoid) of non-negative integers. Let $a<b$ be relatively prime integers such that $2< a$. Let $S :=\mathbb N a +\mathbb N b$ be the semigroup generated ...
- 65
2
votes
1
answer
337
views
Behrend's construction on large 3-AP-free set
Theorem (Behrend's construction)
There exists a constant $C>0$ such that for every positive integer
$N$, there exists a $3$-AP-free $A\subseteq[N]$ with $|A|\geq
Ne^{-C\sqrt{\log N}}$.
Proof. Let $...
- 17k
1
vote
0
answers
115
views
About a proof or disproof of a statement concerning additive bases of natural numbers.
Here $\mathbb{N}=\{1,2,3,\dots\}$.
We say that a set $A\subset\mathbb{N}$ is an additive base of natural numbers if there is
a positive integer $h$ such that every natural number can be written as
$...
5
votes
1
answer
111
views
Is there an examble of a non additive base of natural numbers with ratio of two consecutive terms goes to 1?
Here $\mathbb{N}=\{1,2,3,\dots\}$.
We say that a set $A\subset\mathbb{N}$ is an additive base of natural numbers if there is
a positive integer $h\in \mathbb{N}$ such that every natural number can be ...
2
votes
1
answer
124
views
Roth's Theorem Finitary and Infinitary forms are equal
Theorem1
Let A be a subset of positive integers with positive upper density then A contains a 3 term arithmetic progression.
Theorem2
For any $\delta>0$ there exists $N_{0}$ such that for every $N\...
- 23
1
vote
1
answer
72
views
Upper bound on $\sum_{J\subset [n], |J|=i}\frac{1}{2^{s(J)}}$ where $s(J)=\sum_{j\in J}j$
Define $s(J)=\sum_{j\in J}j$ and $[n]=\{1,...,n\}$. I'm trying to get some reasonable upper bound on $$\sum_{J\subset [n], |J|=i}\frac{1}{2^{s(J)}}.$$
Actually, I want an upper bound on $$\sum_{i=k}^{...
- 1,075
3
votes
1
answer
131
views
Understanding Arithmetic Progression in $[N]$ vs. $\mathbb{Z}_N$
For a set $A$ with some underlying addition operator, $r_k(A)$ is the size of the maximum subset of $A$ that does not contain a $k$-term arithmetic progression.
Exercise 10.0.1 in Tao-Vu's Additive ...
- 158
9
votes
1
answer
154
views
Properties of subsets for which $\sum 1/k$ diverges
The well-known Erdos-Turan conjecture is the following.
Let $V \subset \mathbb{N}$ be such that $\sum_V k^{-1}$ diverges. Then $V$ contains arithmetic progressions of every possible length.
A recent ...
3
votes
1
answer
174
views
Determine the structure of all finite sets $A$ of integers such that $|A| = k$ and $|2A| = 2k + 1$.
An exercise in Nathanson's text: Additive Number Theory, Inverse problems and the geometry of sumsets is the following (Excercise 16, P.No.37):
Determine the structure of all finite sets $A$ of ...
- 622