Related papers: Distinct distances on regular varieties over finit…

We prove that, provided $d > k$, every sufficiently large subset of $\mathbf{F}_q^d$ contains an isometric copy of every $k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices…

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and…

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that…

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…

Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any…

We prove that, under certain conditions on the function pair $\varphi_1$ and $\varphi_2$, bilinear average $p^{-1}\sum_{y\in \mathbb{F}_p}f_1(x+\varphi_1(y)) f_2(x+\varphi_2(y))$ along curve $(\varphi_1, \varphi_2)$ satisfies certain decay…

Let $\mathbb{K}$ be the algebraic closure of a finite field $\mathbb{F}_q$ of odd characteristic $p$. For a positive integer $m$ prime to $p$, let $F=\mathbb{K}(x,y)$ be the transcendency degree $1$ function field defined by…

Let $W$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$ of any characteristic and $mW$ denote the direct sum of $m$ copies of $W$. Let $\mathbb{F}_q[mW]^{{\rm GL}(W)}$ and $\mathbb{F}_q(mW)^{{\rm GL}(W)}$ denote the…

A $k$-uniform, $d$-regular instance of Exact Cover is a family of $m$ sets $F_{n,d,k} = \{ S_j \subseteq \{1,...,n\} \}$, where each subset has size $k$ and each $1 \le i \le n$ is contained in $d$ of the $S_j$. It is satisfiable if there…

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set…

We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then…

We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator $\Delta_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant…

Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the…

Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$ and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a positive-dimensional special subvariety of $X$. We show that the if a coordinate $P_i$…

This paper shows that for K a local field, k a subfield of K and X a variety over k, X is complete if and only if for every finite field extension K' of K, X(K') is compact in its strong topology.

We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.

In these short notes, we will show the following. Let F_q be a finite field and let E/\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E. Under an assumption, we show that it is NP-complete to check if S_r…

We show explicit estimates on the number of $q$--rational points of an $F_q$--definable affine absolutely irreducible variety of the algebraic closure of the finite field $F_q$ of $q$ elements. Our estimates for a hypersurface significantly…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

Given $\{h_1,\cdots,h_{t}\} $ a finite subset of $\mathbb{R}^d$, we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $\mathbb{R}^d$ with the property that the forward differences…