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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).
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Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ over $F$ have rank $leq r$ ? This question is classical, and the answer ($q^{2r}$ when $rleqminleft{ p,qright} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},ldots,x_{k-1}$ for some $kleq rleqminleft{ p,qright} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},ldots,x_{p+q}$ such that the resulting Hankel matrix $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ has rank $leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.
We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.
Given $E subseteq mathbb{F}_q^d times mathbb{F}_q^d$, with the finite field $mathbb{F}_q$ of order $q$ and the integer $d ge 2$, we define the two-parameter distance set as $Delta_{d, d}(E)=left{left(|x_1-y_1|, |x_2-y_2|right) : (x_1,x_2), (y_1,y_2) in E right}$. Birklbauer and Iosevich (2017) proved that if $|E| gg q^{frac{3d+1}{2}}$, then $ |Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| gg q^{frac{10}{3}}$, then $ |Delta_{2, 2}(E)| gg q^2$. In this paper, we present extensions and improvements of these results.
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.
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