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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.
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
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
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
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 …
- 12.3k
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 …
- 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
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
1
vote
0
answers
29
views
Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (wh …
- 10.5k
2
votes
0
answers
137
views
The set of lengths of $nX$ gets larger and larger for every non-zero, non-empty, finite $X \...
Let $H$ be a multiplicatively written monoid with identity $1_H$. Given $x \in H$, we take ${\sf L}_H(x) := \{0\}$ if $x = 1_H$; otherwise, ${\sf L}_H(x)$ is the set of all $k \in \mathbf N^+$ for whi …
- 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
$$
(X, Y) \mapsto X + Y := \{x+y: x \in X \text{ and }y \ …
- 10.5k
6
votes
1
answer
152
views
Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weak...
Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:
(F1) $f( …
CommunityBot
- 1
0
votes
0
answers
125
views
Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a ...
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual …
CommunityBot
- 1
10
votes
1
answer
548
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natura...
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: Convergence …
- 10.5k
2
votes
Accepted
Additivity of upper densities with respect to arithmetic progressions of integers
The answer is in the negative.
Let $f$ and $g$ be two upper densities (in the sense of the OP), and let $\alpha \in [0,1]$ and $q \in [1,\infty[$. Then the function
$$h := (\alpha f^q + (1-\alpha) …
- 10.5k
2
votes
Reference to a variant of Abel's summation formula
Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was ass …
- 10.5k