Questions tagged [sumsets]
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114 questions
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A universal additive complement
Let's say that two subsets of a group are additive complements of each other if their sumset is the whole group. Suppose that the group is finite of prime order. For a fixed $\alpha\in(0,1)$, what is ...
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Unique sums in dilation of sumsets
Let $A \subset [0:d]^n$, then I call $(a,b) \in A^2$ a unique sum $a+b$ cannot be written as $a'+b'$ for some distinct pair $(a',b')$ upto permutation. I conjecture that number of unique sums might be ...
- 341
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Does the sumset inequality $|A+A|\cdot |B+B| \le |A+B|^2$ hold?
The question is more or less the title, though I suppose it is worth mentioning that the energy version of this statement,
$$ E(A,B)^2 \le E(A,A) E(B,B),$$
does in fact hold (it is the Cauchy--Schwarz ...
- 1,251
4
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"Polynomial evaluation estimates" in the spirit of sum-product estimates
A well-studied problem in additive combinatorics is to give sum-product estimates, i.e. lower bounds on $\max\{|A + A|, |AA|\}$ for a set $A \subseteq \mathbb{F}_p$.
I'm interested in a related ...
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Complete subsets in elementary abelian groups
Let's say that a subset $A$ of an abelian group is complete if its subset sum set $\Sigma(A):=\{ \sum_{b\in B} b\colon B\subseteq A, |B|<\infty \}$ is the whole group. Let $\mu(G)$ be the size of ...
- 23.5k
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$AA+AA$ versus $(A+A)(A+A)$
Let $R$ be a commutative ring and let $A\subseteq R$ contain no zero divisors. Under the assumption that $|A+A|\le K|A|$ and $|A\cdot A|\le K|A$, I believe I have a rather simple proof of
$$|(A+A')\...
- 1,251
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Petridis' inequality and Plünnecke's inequality with different summands
One version of the Plünnecke–Ruzsa inequality is the following.
Theorem 1 (Plünnecke–Ruzsa inequality). Let $A, B_1, \ldots, B_m$
be finite subsets of an
abelian group and suppose that $|A+B_i|\le K_i|...
- 1,251
4
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1
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APs in sumsets of exponential growing sequences
I posted this initially on SE, but after I didn't found a particular reference on it, I decided it would be more appropriate to post it here. A friend shared this observation with me and I thought ...
- 83
4
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2
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Lower bounding a sumset quantity
Given $A,B \subset[0,...,d]^n$ such that $A \cap B = \phi$. Can we show
$$ |(2A \cup 2B) \triangle (A + B)| \geq \Omega_d({\rm poly}(|A|,|B|))$$
where $2A = A+A, 2B = B+B$ and we are taking the ...
- 341
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Gaps in sumsets and difference sets
a) Let $S\subset \{1,2,\dotsc,N\}$ be a fairly thick set (with at least $N^{1-\epsilon}$ elements, say). Suppose that the intersection of, say,
$$3 S - 3 S = \{a_1+a_2+a_3-(a_4+a_5+a_6): a_1,\dotsc,...
- 22k
-1
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Is there another representation for the summation: $\sum_{j=1}^{N}\frac{a_j}{(c+a_j)(c+a_j+1)} $, how to reformulate that to keep $c$ out of the sum [closed]
Is there a closed form (without summation) for the summation or at least can I reformulate that so I keep $c$ out of the summation, for example, $c \sum_{n=1}^{N} f(a_n,b_n)$.
$$
\sum_{n=1}^{N}\frac{...
3
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1
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Euro2024-inspired scoring problem
Motivation. The Euro 2024 soccer football championship is in full swing, and the male part of my family are avid watchers. Right now the championship is in the group stage where every group member ...
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235
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Sumsets that contains many squares, Improvement on the bound
I'm being troubled by this problem on AoPS:
https://artofproblemsolving.com/community/c6h1998237p13955033
I searched for any literature related to it such as
Nguyen, Hoi H., and Van H. Vu., Squares ...
- 83
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1
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How to determine if a set is a sumset
Let $G$ be a commutative group (assume whatever you want on $G$ if needed. I am mainly interested in $G = \mathbb{Z}/n\mathbb{Z}$).
Let $k$ be a fixed integer.
Let $(a_1, \dots, a_{k^2})$ be a list of ...
- 537
6
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Normality and small doubling
Suppose that $A$ is a finite, generating subset of a group $G$, and that $H$ is a subgroup such that $A^2$ is a union of left $H$-cosets; moreover, $H$ is maximal subject to this property. Is it true ...
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