All Questions
8 questions with no upvoted or accepted answers
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
4
votes
0
answers
104
views
Visiting zero-sum triples in a vector space
Is it true that for any set $A\subset\mathbb F_5^n$ satisfying $A\cap(-A)=\varnothing$, there is a subset $A'\subseteq A$ such any triple $(a,b,c)\in A\times A\times A$ with $a+b+c=0$ has exactly one ...
- 23k
3
votes
0
answers
293
views
Approximate versions of Segre's Theorem
Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...
- 13k
3
votes
0
answers
147
views
Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?
Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$.
Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...
3
votes
0
answers
181
views
On an exercise in The Probabilistic Method : random dilate of a set in a finite field
This is related to Problem $4.6$ in ``The Probabilistic Method'' by Alon and Spencer, where one essentially has to prove the following:
Let $p$ be a prime, and $A$ be any subset of $\mathbb{F}_p$. ...
- 31
3
votes
0
answers
73
views
Number of classes $\pmod p$ represented by $b_1s^{n-1} + \dots + b_n$ where $ord_p(s) = n$
Let $n \in \mathbb Z$ with $n \ge 3$ and let $p$ be a prime number such that $n|p-1$. Let $a_1,a_2,\dots,a_{2n-1} \in \mathbb Z/p\mathbb Z$. Suppose that the same class is represented by at most $n-1$ ...
- 169
2
votes
0
answers
187
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
1
vote
0
answers
154
views
On the set $\{\sum_{k=1}^n \lambda_ka_k:\ a_1,\ldots,a_k\ \text{are distinct elements of}\ A\}$
For a field $F$ let $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\infty$ if $F$ is of characteristic zero.
In 2007 I considered the linear extension of the Erdos-Heilbronn ...
- 15.6k