All Questions
8 questions with no upvoted or accepted answers
11
votes
0
answers
830
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 ...
- 23k
9
votes
0
answers
265
views
If $A+A+A$ contains the extremes, does it contain the middle?
Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers).
Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \...
- 825
7
votes
0
answers
177
views
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 ...
- 73
5
votes
0
answers
185
views
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,
$$3 S - 3 S = \{a_1+a_2+a_3-(a_4+a_5+a_6): a_1,\dotsc,...
- 20.2k
4
votes
0
answers
216
views
Subgroup cliques in the Paley graph
It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...
- 23k
2
votes
0
answers
137
views
The set of lengths of $nX$ gets larger and larger for every non-zero, non-empty, finite $X \subseteq \mathbf N$ with $0 \in X$
Let $H$ be a multiplicatively written monoid with identity $1_H$. Given $x \in H$, we take ${\sf L}_H(x) := \{0\}$ if $x = 1_H$; otherwise, ${\sf L}_H(x)$ is the set of all $k \in \mathbf N^+$ for ...
- 10.5k
0
votes
0
answers
87
views
Asymptotic behavior of sumsets of squares with restricted congruence conditions
Recall that if $A$ and $B$ are both subsets of the integers, then $A+B=\{a+b:a \in A,b \in B\}$.
Lagrange's four-square theorem states that if $A$ is the set of squares, then $4A=A+A+A+A=\mathbb{N}$.
...
- 101
0
votes
0
answers
167
views
General asymptotic result in additive combinatorics (sums of sets)
Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...
- 3,098