Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with additive-combinatorics
Search options not deleted
user 16537
Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the top-level tags nt.number-theory or co.combinatorics. Some additional tags are available for further specialization, including the tags sumsets and sidon-sets.
6
votes
1
answer
152
views
Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weak...
Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:
(F1) $f( …
- 10.5k
2
votes
Accepted
Additivity of upper densities with respect to arithmetic progressions of integers
The answer is in the negative.
Let $f$ and $g$ be two upper densities (in the sense of the OP), and let $\alpha \in [0,1]$ and $q \in [1,\infty[$. Then the function
$$h := (\alpha f^q + (1-\alpha) …
- 10.5k
3
votes
0
answers
111
views
Decomposing a subset of $\mathbf Z$ into a sumset of irreducibles
We say that a subset $A$ of $\mathbf Z$ is irreducible if $|A| \ge 2$ and there do not exist $X, Y \subseteq \mathbf Z$ with $|X|, |Y| \ge 2$ such that $A = X + Y$.
If $X \subseteq \mathbf Z$, we de …
- 10.5k
18
votes
Accepted
Sets that are not sum of subsets
There are three questions in the OP, and I'll try to address each of them in as comprehensive a way as I can. I'll be glad to add in further details (and references) if requested.
◇ Preliminaries on f …
- 10.5k
2
votes
1
answer
578
views
On the upper Banach density of the set of positive integers whose base-$b$ representation mi...
Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A …
- 10.5k
0
votes
0
answers
125
views
Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a ...
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual …
- 10.5k
3
votes
0
answers
96
views
Given a primitive finite set $A\subseteq\bf N$ with $0\in A$, find two more primitive sets $...
Let $\mathcal P_{{\rm fin},0}(\mathbf N)$ be the monoid of all finite subsets of $\mathbf N$ containing $0$ with the operation of set addition
$$
(X, Y) \mapsto X + Y := \{x+y: x \in X \text{ and }y \ …
- 10.5k
2
votes
Conditions for an analogue of Cauchy-Davenport for simple groups
Not sure whether @David is still around here, but I'd like to add a complement to @quid's answer.
Fix an integer $n \ge 9$, and let $q$ be a prime power and $\mathbb G = (G, \cdot)$ the projective s …
- 10.5k
3
votes
1
answer
514
views
Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the co …
- 10.5k
9
votes
1
answer
408
views
On the origin of a fundamental theorem of additive number theory
Given $a, b \in \mathbb Z$, set $[\![a,b]\!] := \{x \in \mathbb Z: a \le x \le b\}$. A basic result in additive number theory goes as follows:
If $A$ is a finite subset of $\mathbb N$ with $0 \in A$ …
- 10.5k
12
votes
Accepted
On the origin of a fundamental theorem of additive number theory
I shared the link to this thread with several colleagues, inviting them to contribute to the discussion. Notably, I received a response from Melvyn Nathanson himself, most of which is reproduced below …
- 10.5k
4
votes
Accepted
Maximal zero-sum free sequences of $C_3^n$
I sent an email to Alfred Geroldinger with a link to this thread. Here is a summary of his reply (I'm posting with his permission):
The structure of minimal zero-sum sequences of maximal length over …
- 10.5k
3
votes
2
answers
614
views
Who needs a symmetric upper asymptotic density on the integers?
The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric varia …
- 10.5k
1
vote
Who needs a symmetric upper asymptotic density on the integers?
Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.
1. A reference to t …
- 10.5k
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if...
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the upp …
- 10.5k