Questions tagged [additive-combinatorics]
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.
665
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Unique "clique" of differences in $\mathbb{Z}/m\mathbb{Z}$
Are there absolute constants $0 < \epsilon < 1$ and $N \in \mathbb{N}$ such that the following holds: For every $m \in \mathbb{N}$ and every $A \subseteq \mathbb{Z}/m\mathbb{Z}$ with $\frac{\...
- 73
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Szemeredi Regularity Lemma - Reasonable Bounds
Recently, I came across the wonderful Szemeredi regularity lemma and I was wondering. Are these "natural/reasonable/standard" examples of random graphs families, for which the number of $\...
- 4,969
2
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0
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258
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On $(k,\ell)$-sumfree sets
Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation
$$x_1+\dots +x_k = y_1+\dots +y_\ell$$
in the set (for distinct $x_i$'s and $...
- 781
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0
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Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
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4
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Binomial coefficient in Andrews' partition book
First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and ...
- 4,618
5
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2
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553
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Representing natural numbers as sums of distinct prime powers
I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
- 99.1k
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1
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Bounding maximum sum of integer matrix entries in a non-attacking rook placement
Let $A =(a_{ij})$ be a $m \times n$ matrix with nonnegative integer entries bounded above by $k$. To find the set of entries of $A$ in a non-attacking rook placement such that the sum $S$ of them is ...
- 5,169
2
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1
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196
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Double estimates relating Ruzsa distance and doubling constant
I am trying to solve the following exercise (2.3.16) from Tao-Vu book.
Let $A,B$ be additive sets with common ambient group $Z$. Show that
$\sigma[A\cup B]\leq e^{d(A,B)}+2e^{4d(A,B)}$. In the ...
CommunityBot
- 1
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0
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74
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Number of solutions $x$ of equation $a_1 b_1^x + \dotsb + a_n b_n^x=0$ over a finite field
Let $F$ be a finite field and let $a_1, b_1, \dotsc, a_n, b_n \in F$ be field elements. I am interested in the number of solutions $0\leq x \leq |F|-1$ such that
\begin{equation}\label{e:1}
a_1 b_1^x +...
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1
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Density of a set of numbers whose prime factors are defined by congruences
Let $S$ be the set of positive integers not divisible by $3$ where if $p$ is a prime factor of $n \in S$ and $p \equiv 1\bmod 3$ then $p^2$ does not divide $n$, but if $p\equiv2 \bmod 3$ then $p^2$ ...
- 1,191
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1
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Expected number of coin flips before you see a $k$-term arithmetic progression of heads
Let $\{X_i\}_{i \in \mathbb Z_+} $ be independent fair coin flips. Write $S := \{i \in \mathbb Z_+\, | \, X_i \text{ is heads}\}$, and define, for an integer $k \geq 3$,
$$Y := \inf \{n \in \mathbb N \...
- 7,434
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0
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75
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High probability bound on number of sparse solutions to Gaussian linear system
Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...
- 43
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1
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Khovanskii's theorem on iterated sumsets
I was watching Gowers video lectures "Introduction to Additive Combinatorics" (my question is about the statement he made at the 21st minute) and came across wonderful theorem due to ...
CommunityBot
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2
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Acyclic matching property in $\mathbb{Z}/p\mathbb{Z}$
Let $B$ be a finite subset of the group $G$ which does not contain the neutral element. For any subset $A$ in $G$ with the same cardinality as $B$, a matching from $A$ to $B$ is defined to be a ...
- 6,571
1
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1
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Instance of polynomial van der Waerden without good bounds
Let $P\subset \Bbb{Z}[X]$ be a finite set of polynomials with constant-term zero. Then, polynomial vdW says:
For eacg finite $r$, there exists some $N=N(P,r)$, such that every $r$-coloring $C:\{1,\...
- 383
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0
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On fifth powers forming a Sidon set
We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...
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Is every real number in [0,1] a product of three (or more) Cantor set's numbers?
It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
- 2,427
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0
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Upper bound of number of different rows for a binary matrix
Let $\mathcal{X} \subseteq \mathbb{R}^n$ be a measurable space, Fix $m,q \in \mathbb{N}^*$, (the $m$ $x_i$'s are i.i.d and follow distribution $\mathcal{D}$)
and $X = (x_1, \dots, x_m) \sim \mathcal{D}...
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Are there a few input bits that randomize the output of an $\mathbb{F}_2$ polynomial?
Suppose $f:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$ is a degree $d$ polynomial and $\epsilon>0$ is some real number. Does there necessarily exist a set $C\subset [n]$ of coordinates with the size of ...
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Growth of powers of symmetric subsets in a finite group
(This question was originally asked on Math.SE, where it was answered in the abelian case)
Let $G$ be a finite group, and let $A$ be a symmetric subset of $G$ containing the identity (i.e., $A^{-1}=A$ ...
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2
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The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 1,193
2
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2
answers
339
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Why can we not find exact values for sizes of cap sets for $d>6$?
I've been reading about cap sets in $\mathbb{F}_3^d $ over the past couple of days and wondered why we can only find bounds, as opposed to exact values, for (maximum) sizes of cap sets for $d>6$. ...
- 1,603
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2
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What is the minimal density of a set A such that A+A = N?
Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$?
What I know:
...
- 3,964
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2
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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 $...
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0
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subsets of $\mathbb{N}$ whose shifts have finite intersection property in density
I am interested in proving the statement:
Let $S\subseteq\mathbb{N}$ such that for every $r\in\mathbb{N}$ and for every $k_{1}$, $k_{2}$, $\ldots$, $k_{r}\in\mathbb{N}$, the set $\big(S-k_{1}\big) \...
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A question about Erdős thin bases
Let $B \subset \mathbb{N}$ be an additive basis of order $h$. Define $r_{B,h}(n)$ to be the number of ways $n$ can be written as a sum of $h$ elements of $B$. In particular, $B$ is a basis of order $h$...
- 3,964
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1
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Structural description of Bohr sets in $\mathbb{Z}_N$
Definition 1. Let $\Gamma\subset \widehat{G}$ and $\delta\in [0,2]$. The Bohr set with frequency set $\Gamma$ and width $\delta$ is the set $\text{Bohr}(\Gamma; \delta)= \big\{x\in G: |\chi(x)-1|\leq \...
- 22.8k
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When can $L$ sets of the form $\{a,b,a+b\}$ partition $\{1,2,\dots, 3L\}$?
Firstly, this question has been posted to Math StackExchange with no complete answer so far.
Consider a set of the form $\{a,b,a+b\}$ where $a$ and $b$ are positive integers with $b > a$. I will ...
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1
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Corners theorem in finite fields
The corners theorem of Ajtai and Szemerédi states that if $A\subseteq[N]^2$ is corner-free, i.e. there are no $x,y,h\in\mathbb{N}$ with all of $(x,y),(x+h,y),(x,y+h)$ in $A$, then $|A|=o(N^2)$. The ...
- 11.3k
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Is number of different sums monotone?
Suppose you have a set $S$ consisting of $n$ different integers.
Let $$W_k = \#\biggl\{x\in\Bbb Z\colon \text{there exists } T \subseteq S,\, \#T=k,\, \sum_{a \in T} a = x\biggr\}.$$
My question is: ...
- 109k
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1
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Upper bounds estimates of Minkowski sum
Let $A,B \subset \{0,...,d\}^n$, do we have any result that says $|A+B| \leq \mathcal{O}_d(|A|\cdot |B|)^\tau$ for $\tau < 1$. The case $\tau = 1$ is trivial, and due to the restricted setting, I ...
- 3,413
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0
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On two formulas involving the $k$-fold divisor function $d_k$ and the function $r_k$
I have a puzzle which needs some help form the experts here. Let $d(n)$ be the divisor function, and $d_k(n)$ the $k$-fold divisor function.
I) It is known that, for any positive integer $h$,
$$d(n+h)...
- 10.1k
5
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1
answer
901
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Estimate of Minkowski sum
Let A $\subset [0:2]^n$, where $[0:2]=\{0,1,2\}$, then define $2A= \{ a+b\mid a,b \in A \}$. I wanted to know the best known lower-bound estimates for $|2A|$.
I intuitively expect that $|2A| \geq |A|^{...
- 11.9k
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1
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On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
CommunityBot
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Subsets of $(\mathbb{Z}/p)^{\times n}$
There seems to be some combinatorial fact that every subset $A$ of $G=(\mathbb{Z}/p)^{\times n}$ of cardinality $\frac{p^n-1}{p-1}+1$ containing $\vec{0}$ satisfies $(p-1)A=G$. ($p$ is a prime number....
- 85.1k
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1
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Primitive recursive bounds for multidimensional polynomial vdW / HJ
In Shelah's paper 679, he proves primitive recursive bounds for the polynomial Hales-Jewett theorem and thus for the polynomial van der Waerden theorem.
How about for the multidimensional polynomial ...
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0
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The number of incidences between points and parabolas on $\mathbb{R}^2$
I was reading Adam Sheffer's book "Polynomial Methods and Incidence Theory" and I tried to solve the following exercise:
Exercise 1.1 Construct a set $\mathcal{P}$ of $m$ points and a set $\...
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1
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659
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A variant of the corners problem
Question: What is the size of the largest subset of $[n]^2$ containing no three point configurations of the form $(x,y), (x,y+d), (x+d,y')$ with $d \neq 0$? In particular, is it at most $O(n)$?
Recall ...
- 103k
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1
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Beating trivial bound for $k$-AP-free sets in characteristic $k$
Given integers $k,n\ge 1$, I shall write $\Bbb{Z}_k^n := (\Bbb{Z}/k\Bbb{Z})^n$.
Fix $k\ge 3$. Let $r_k(\Bbb{Z}_k^n)$ denote the cardinality of the largest $A\subset \Bbb{Z}_k^n$, such that $A$ does ...
- 18.4k
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0
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Equivalent formulation of Szemerédi-Trotter theorem
I am reading the first chapter of Adam Sheffer's book "Polynomial Methods and Incidence Theory" and in Lemma 1.15 he proves an equivalent formulation of Szemerédi-Trotter theorem. Before ...
- 298
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1
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A summation involving fraction of binomial coefficients
I need to prove the following statement.
Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $ ...
- 30.6k
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1
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Szemerédi–Trotter type theorem in finite field
This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao.
In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known
$$|A''+A''|\lesssim ...
- 11.8k
1
vote
0
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54
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Largest interval containing family of sets with an overlap property
Here's a simplified version of a question I'm interested in.
Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly ...
- 539
2
votes
1
answer
109
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Capset problem but considering differences with bounded support
For any dimension $D\ge 1$, we define the homomorphism $\phi: \Bbb{Z}^D\to (\Bbb{Z}/3\Bbb{Z})^D; \xi\mapsto \xi+3\Bbb{Z}^D$.
Given a set $A \subset \{0,1,2\}^D$, we define $S_A$ to be the set of $v \...
- 3,413
1
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0
answers
68
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Progressions in finite fields with bounded hamming weight
Given $k\ge 2$ and an additive set $S$ (understood to live some implicit group $G$), define $$\Delta_k(S) := \left\{ d \in G: \bigcap_{i=1}^k (S+i\cdot d) \neq \emptyset \right\} $$(i.e., this is the ...
- 3,413
2
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0
answers
173
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Component-wise sums of permutations
Given a set $S$ containing all possible permutations of a vector $v = (1, 2, 3, ..., n-1, n)$, find the size of the set $P$, where $P$ is defined as the set of possible component-wise sums obtained by ...
- 21
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0
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120
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An additive combinatoric probability question
I asked the question in cs-theory stack exchange but was advised a pure math forum would be more apt. Link to the question:
https://cstheory.stackexchange.com/questions/52930/an-additive-combinatoric-...
- 159
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2
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For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
I am currently working on a proof that would need to use the following theorem that I cannot prove:
"Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least ...
- 633
6
votes
0
answers
491
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At most two elements give 1 to n
Fix a positive integer $m$. Let $n$ ( $= n(m)$) be the largest positive integer for which there exists some subset $\{a_1,\ldots,a_m\} \subseteq \{1,2,\ldots,n\}$ of $m$ positive integers between $1$ ...
- 165
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0
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128
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Counting $A+A-A$ with partial multiplicity
A recent question asked whether, given a finite set of positive numbers $A$, it is always the case that the set $A+A-A$ has more positive than negative elements. Terry Tao showed that this is false (...
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