3
votes
0answers
120 views

Subgroup cliques in the Paley graph

It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...
3
votes
3answers
300 views

How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different. Example with $|A|=3$: Out of the set $A ...
1
vote
0answers
36 views

$B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds $$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$ Thus if you know the sum of two elements, you know which elements ...
23
votes
1answer
455 views

Distribution of $a^2+\alpha b^2$

It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...
9
votes
1answer
358 views

Lower bounds for $|A+A|$ if $A$ contains only perfect squares

Let $A$ a set with $|A|=n$ that contains only perfect squares of integers. What lower bounds can we give for $|A+A|$? I think the lower bound $\gg \frac{n^2}{\sqrt{log \,n}}$ holds (this would be ...
4
votes
1answer
338 views

A problem related with 'Postage stamp problem'

A friend of mine taught me this question. I found that it is related with 'Postage stamp problem' (though it does not seem to be same). Let $m,a_1\lt a_2\lt \cdots\lt a_n$ be natural numbers. Now let ...
1
vote
1answer
151 views

unique sums in a finite direct product of sets of integers

I am an algebraist, and I am wondering if there is a definition for the following: Let $A_1$, $A_2$, $\ldots, A_n$ be sets of integers (or more generally, subsets of a group $G$). Say that (for the ...
9
votes
0answers
513 views

Cliques in the Paley graph and a problem of Sarkozy

The following question is motivated by pure curiosity; it is not a part of any research project and I do not have any applications. The question comes as an interpolation between two notoriously ...