Related papers: An improved point-line incidence bound over arbitr…
Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this…
In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P}, \mathcal{V})$ between a set $\mathcal{P}$…
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in…
In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These…
The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More…
The point-line incidence problem has been widely studied in Euclidean spaces and vector spaces over finite fields, whereas the analogous problem has rarely been considered over finite $p$-adic rings. In this paper, we investigate incidences…
We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence results, they outperform their more general counterparts in…
In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the…
We improve the well-known Szemer\'edi-Trotter incidence bound for proper 3--dimensional point sets (defined appropriately)
In this paper we prove an incidence bound for points and cubic curves over prime fields. The methods generalise those used by Mohammadi, Pham, and Warren (2021).
We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds…
In this short note, we use Rudnev's point-plane incidence bound to improve some results on conditional expanding bounds for two-variable functions over arbitrary fields due to Hegyv\'ari and Hennecart.
In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a…
The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as…
We give a new proof of VC bounds where we avoid the use of symmetrization and use a shadow sample of arbitrary size. We also improve on the variance term. This results in better constants, as shown on numerical examples. Moreover our bounds…
We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the…
We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of…
The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of…
We establish improved finite field Szemeredi-Trotter and Beck type theorems. First we show that if P and L are a set of points and lines respectively in the plane F_p^2, with |P|,|L| \leq N and N<p, then there are at most C_1…
We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain-Katz-Tao finite field incidence theorem derived from sum-product estimates. In…