We prove a new lower bound for the number of pinned distances over finite fields: if $A$ is a sufficiently small subset of $\mathbb{F}_q^2$, then there is an element in $A$ that determines $\gg |A|^{2/3}$ distinct distances to other elements of $A$. Combined with results for large subsets $A\subseteq\mathbb{F}_q^2$, this improves all previously known lower bounds on distinct distances over finite fields.
In fact, we obtain an upper bound for the number of isosceles triangles determined by $A$. For that we use the concept of bisector energy. It turns out that the latter can be expressed as a point-plane incidence bound, so one can use a theorem of the third author.
The conversion to this incidence problem relies on the Blaschke-Grünwald kinematic mapping -- an embedding of the group of rigid motions of $\mathbb{F}_q^2$ into an open subset of the projective three space. This has long been known in kinematics and geometric algebra; we provide a proof for arbitrary fields using Clifford algebras.
We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their bisector energy.
Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $Ω(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive proportion of points $a \in P$, we have $$|\{\| a- b\|: b \in P\}|=Ω(q),$$ where $\|a- b\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).
We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.
Let $R$ be a finite valuation ring of order $q^r$ where $q$ is odd and $A$ be a subset of $R$. In the present paper, we prove that there exists a point $u$ in the Cartesian product set $A\times A\subset R^2$ such that the size of the pinned distance set at $u$ satisfies $$|Δ_u(A\times A)|\gg \min\left\{q^r, \frac{|A|^3}{q^{2r-1}}\right\}.$$ This implies that if $|A|\ge q^{r-\frac{1}{3}}$, then the set $A\times A$ determines a positive proportion of all possible distances.
We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.
We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer.
In this paper we obtain improved dimensional thresholds for dot product sets corresponding to compact subsets of a paraboloid. As a direct application of these estimates, we obtain significant improvements to the best known dimensional thresholds that guarantee that a given compact subset of Euclidean space determines a positive proportion of all possible congruence classes of simplexes. In many regimes this improves the results previously obtained by Erdogan-Hart-Iosevich (\cite{EHI}), Greenleaf-Iosevich-Liu-Palsson (\cite{GILP}) and others.
We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$Δ^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a probability measure $μ_F$ on $F$ such that for $μ_F$-a.e. $y\in F$, (1) $\dim_{\mathcal H}(Δ^y(E))\geqβ$ if $\dim_{\mathcal H}(E) + \frac{d-1}{d+1}\dim_{\mathcal H}(F) > d - 1 + β$; (2) $Δ^y(E)$ has positive Lebesgue measure if $\dim_{\mathcal H}(E)+\frac{d-1}{d+1}\dim_{\mathcal H}(F) > d$; (3) $Δ^y(E)$ has non-empty interior if $\dim_{\mathcal H}(E)+\frac{d-1}{d+1…
An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff. The pinned distance set $Δ_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in E\}$ for a pin $y\in E$ has…
The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set $Δ(A)$ has positive Lebesgue measure. In a very recent paper, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for $E\subset\mathbb{F}_p^2$ with $|E|\gg p^{5/4}$, there exist many points $x\in E$ such that the number of distinct distances from $x$ is at least $cp$. The main purpose of this paper is to pr…
In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this result, as well as a stronger "pinned variant", was given by Bourgain in \cite{B} using Fourier analytic methods. In \cite{M1} the second author adapted Bourgain's Fourier analytic approach to established a result analogous to that of Katznelson and Weiss for subsets $\mathbb{Z}^d$ provided $d\geq 5$. We present a new direct proof of this discrete distance s…
Given $E \subset \mathbb{F}_q^d$, we show that certain configurations occur frequently when $E$ is of sufficiently large cardinality. Specifically, we show that we achieve the statistically number of $k$-stars $\displaystyle\left|\left\{(x, x^1, \dots, x^k) \in E^{k+1} : \| x - x^i \| = t_i \right\}\right|$ when is $|E| \gg_k q^{\frac{d+1}{2}}$. This result can be thought of as a natural generalization of the Erd\H os-Falconer distance problem. Our result improves on a pinned-version of our theorem which implied the above result, but only in the range $|E| \gg q^{\frac{d+k}{2}}$. As an immedia…
We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. These estimates stem from specially suited applications of incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios defined by triples of elements of a (sufficiently small if $\mathbb{F}$ has…
