Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFurstenberg sets in finite fields: Explaining and improving the Ellenberg-Erman proof
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A subset $S subset mathbb{F}_q^n$, where $mathbb{F}_q$ is a finite field, is called $(k,m)$-Furstenberg if it has $m$ common points with a $k$-flat in each direction. That is, any $k$-dimensional subspace of $mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Using sophisticated scheme-theoretic machinery, Ellenberg and Erman proved that $(k,m)$-Furstenberg sets must have size at least $C_{n,k}m^{n/k}$ with a constant $C_{n,k}$ depending only $n$ and $k$. In this work we follow the overall proof strategy of Ellenberg-Erman, replacing the scheme-theoretic language with more elementary machinery. In addition to presenting the proof in a self-contained and accessible form, we are also able to improve the constant $C_{n,k}$ by modifying certain key parts of the argument.
rate research
Read More
A $(k,m)$-Furstenberg set $S subset mathbb{F}_q^n$ over a finite field is a set that has at least $m$ points in common with a $k$-flat in every direction. The question of determining the smallest size of such sets is a natural generalization of the finite field Kakeya problem. The only previously known bound for these sets is due to Ellenberg-Erman and requires sophisticated machinery from algebraic geometry. In this work we give new, completely elementary and simple, proofs which significantly improve the known bounds. Our main result relies on an equivalent formulation of the problem using the notion of min-entropy, which could be of independent interest.
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $log r / log s$ is irrational and $X$ and $Y$ are $times r$- and $times s$-invariant subsets of $[0,1]$, respectively, then $dim_text{H} (X+Y) = min ( 1, dim_text{H} X + dim_text{H} Y)$. Our main result yields information on the size of the sumset $lambda X + eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.
Let $mathbb{F}_q$ be a finite field of order $q$, and $P$ be the paraboloid in $mathbb{F}_q^3$ defined by the equation $z=x^2+y^2$. A tuple $(a, b, c, d)in P^4$ is called a non-trivial energy tuple if $a+b=c+d$ and $a, b, c, d$ are distinct. For $Xsubset P$, let $mathcal{E}^+(X)$ be the number of non-trivial energy tuples in $X$. It was proved recently by Lewko (2020) that $mathcal{E}^+(X)ll |X|^{frac{99}{41}}$ for $|X|ll q^{frac{26}{21}}$. The main purposes of this paper are to prove lower bounds of $mathcal{E}^+(X)$ and to study related questions by using combinatorial arguments and a weak hypergraph regularity lemma developed recently by Lyall and Magyar (2020).
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $mathbb{F}_q^2$ over a finite field $mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $mathbb{F}_q$. It turns out that integral point sets over $mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $mathbb{F}_q$ for $qle 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.
In this paper, we study dot-product sets and $k$-simplices in vector spaces over finite rings. We show that if $E$ is sufficiently large then the dot-product set of $E$ covers the whole ring. In higher dimensional cases, if $E$ is sufficiently large then the set of simplices and the set of dot-product simplices determined by $E$, up to congurence, have positive densities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
