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Results tagged with additive-combinatorics
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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.
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
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
10
votes
1
answer
548
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natura...
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: Convergence …
- 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
7
votes
1
answer
480
views
Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mat...
ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ f …
- 10.5k
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
5
votes
0
answers
79
views
Some questions about the Lévy monoid of certain densities
Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: …
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5
votes
1
answer
227
views
Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathb...
Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X …
- 10.5k
4
votes
2
answers
666
views
Reference to a variant of Abel's summation formula
Edit. A stronger version of the formula is true (details follow).
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that …
- 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 …
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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
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 …
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3
votes
0
answers
131
views
Reference for a lemma on the asymptotic upper density of special sets with large gaps and in...
Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am writin …
- 10.5k
3
votes
1
answer
179
views
Decomposing a set of integers as a union of well-separated (discrete) intervals
Let a discrete interval be a set of the form $\{x \in \mathbb Z \colon a \le x \le b\}$ with $a, b \in \mathbb Z \cup \{\pm \infty\}$. Then define the boxing dimension $\text{bim}(S)$ of a set $S \sub …
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