All Questions
Tagged with sumset elementary-number-theory
6
questions
2
votes
1
answer
111
views
Maximal size of bounded “sparse” sets of natural numbers
Let’s call $A \subset \mathbb{N}$ sparse iff for all quadruples of distinct numbers $(a, b, c, d)$ from $A$ it is true, that $a + b \neq c + d$. What is the maximal possible size of a sparse set $A$, ...
- 15.7k
2
votes
2
answers
98
views
Show that there exists $i\in \lbrace 1, 2, 3 \rbrace $ s.t. there exists $a, b\in A_i $ s.t. $a+b\in B $.
Let $A=\lbrace 1, 2, 3,..., 2019\rbrace= A_1\cup A_2\cup A_3$, where $A_1\cap A_2=A_2\cap A_3= A_1\cap A_3=\emptyset $ and $B=\lbrace 672, 1008, 1344, 1680, 2016\rbrace $.
Show that there exists $i\...
- 2,095
1
vote
0
answers
47
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sigma notation for sumsets $\Sigma B$
I am reading a number theory paper about sumsets [1]. If $A, B \in \mathbb{Z}$ are two sets of integers we can define:
$$ A + B = \{ a + b : a \in A, b \in B \} \subseteq \mathbb{Z}$$
what do you ...
- 24.5k
1
vote
0
answers
42
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$|2C|<2^k|C|$ if $C$ is a generalized arithmetic progression
Let $C$ be a generalized arithmetic progression, i.e., given positive integers $N_1,\ldots,N_k$ and $a_0,a_1,\ldots,a_k$ then
$$
C=\left\{a_0+\sum_{i=1}^n a_in_i: 0\le n_i \le N_i-1 \text{ for all }i=...
- 15.5k
2
votes
1
answer
229
views
Trying to understand an assumption in the proof of Mann's Theorem
I am trying to follow the reasoning in the proof of Mann's Theorem:
$$d(C) \ge \min(d(A)+d(B),1)$$
I am clear that we can assume that:
$d(A) + d(B) \le 1$
We only need to prove that for every $n \ge ...
- 10.5k
1
vote
1
answer
101
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Question about Schnirelmann Density and Sumset: if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, wouldn't $d(A+B)=1$
I've been thinking about the Schnirelmann Density and I think that I may still be confused about SumSet and Density.
It seems to me that if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, then $d(A+{B}) = ...
- 10.5k