All Questions
37
questions
19
votes
2
answers
645
views
Sumset that covers $\mathbb{Z}/p\mathbb{Z}$.
Let $p$ be a prime. Let $S$ be a set of residues modulo $p$. Define
$$S^2 = \{a \cdot b \mid a \in S, b \in S\}.$$
Question:
How small can we make $|S|$ such that $\{0, 1, \cdots, p-2, p-1\} \in ...
- 8,165
9
votes
1
answer
154
views
Properties of subsets for which $\sum 1/k$ diverges
The well-known Erdos-Turan conjecture is the following.
Let $V \subset \mathbb{N}$ be such that $\sum_V k^{-1}$ diverges. Then $V$ contains arithmetic progressions of every possible length.
A recent ...
8
votes
2
answers
434
views
How to show that $\gcd(a_1,a_2,\cdots,a_k) = 1$ implies that there exist a non-negative solution to $\sum_{i=1}^{n}a_ix_i = n$ for large $n.$
I was reading about the Coin-problem and I am unable to fully understand the following argument:
On the other hand, whenever the GCD equals 1, the set of integers that cannot be expressed as a ...
- 9,258
8
votes
1
answer
164
views
An analogue of direct sum decomposition of a cofinite subset of integers
Let $S \subseteq \mathbb Z$ be such that $\mathbb Z \setminus S$ is finite , then is it true that there exists infinite $ S_1,S_2 \subseteq \mathbb Z$ such that $S_1+S_2=S$ and for every $s \in S $ , ...
user228169
6
votes
2
answers
640
views
Number-Theoretic Coin Puzzle
There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...
- 899
6
votes
1
answer
403
views
Show there’s at most $n\choose \left \lfloor\frac{n}{2} \right\rfloor$ subsets $A\subset[n]$ such that $\displaystyle\sum\limits_{i\in{A}} a_i=\alpha$
Let $a_1, a_2, a_3, ... , a_n$ and $\alpha$ be n+1 non-zero real numbers. Prove that there are at most $n\choose \left\lfloor\frac{n}{2}\right\rfloor$ subsets $A\subset[n]$ such that $\displaystyle\...
- 195
6
votes
0
answers
131
views
Odd of the form $a^2+b^2+c^2+ab+ac+bc$.
The computation below shows that (for $a,b,c \in \mathbb{N}$) the form $$a^2+b^2+c^2+ab+ac+bc$$ covers every odd integer less than $10^5$ except those in $$I= \{ 5, 15, 23, 29, 41, 53, 59, 65, ...
- 5,847
5
votes
1
answer
460
views
Arithmetic progressions
What are the largest known lower bounds for $B_k$, the maximal sum of the reciprocals of the members of subsets of the positive integers which contain no arithmetic progressions of length $k$?
for $k=...
- 53
5
votes
2
answers
358
views
Is there any formula for this sum of power of positive integers? [duplicate]
I wonder if there is any formula for this sum.
$$k^\gamma+(k-1)^\gamma+\cdots+1^\gamma,$$
where $k$ is positive integer and $\gamma\in(0,1)$. And how about $\gamma<0$? Or is there any known ...
- 2,075
5
votes
1
answer
111
views
Is there an examble of a non additive base of natural numbers with ratio of two consecutive terms goes to 1?
Here $\mathbb{N}=\{1,2,3,\dots\}$.
We say that a set $A\subset\mathbb{N}$ is an additive base of natural numbers if there is
a positive integer $h\in \mathbb{N}$ such that every natural number can be ...
4
votes
1
answer
326
views
Application of Cauchy-Davenport
Let $ p $ be a prime number and $ A \subset \mathbb{Z}/p\mathbb{Z} $. Suppose $ 0 \notin A $ and for $ a \in A $ define $ d(a)= \min\{k|-a \in \underbrace{A+A+ \dots +A}_\text{k times} \} $. I want ...
- 1,085
3
votes
1
answer
176
views
Determine the structure of all finite sets $A$ of integers such that $|A| = k$ and $|2A| = 2k + 1$.
An exercise in Nathanson's text: Additive Number Theory, Inverse problems and the geometry of sumsets is the following (Excercise 16, P.No.37):
Determine the structure of all finite sets $A$ of ...
- 622
3
votes
1
answer
132
views
Understanding Arithmetic Progression in $[N]$ vs. $\mathbb{Z}_N$
For a set $A$ with some underlying addition operator, $r_k(A)$ is the size of the maximum subset of $A$ that does not contain a $k$-term arithmetic progression.
Exercise 10.0.1 in Tao-Vu's Additive ...
- 158
3
votes
0
answers
65
views
Product set in a finite field
For a finite nonempty subset $A$ of a ring $X=(X,+,\cdot)$, let us denote the set $\{a \cdot b \colon a, b \in X\}$.
If $X=\mathbb{Z}$, it is not difficult to show that
$$|AA| \leq \frac{|A|^{2}+|...
- 577
3
votes
0
answers
65
views
Maximal number of subsets of $n$ real numbers that have the same sum, when $2^{n-2}$ of subsets have "unique" sums
I've recently found a problem that I still can't solve: Dependency of the properties of numbers' subsets concerning subsets' sums of $n$ real numbers.
A problem linked with it, that may be ...
- 41