Related papers: A modular Szemeredi-Trotter theorem for hyperbolas

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of…

We describe the $p$-divisibility transposition for the Fourier coefficients of Hermitian modular forms. The results show that the same phenomenon as that for Siegel modular forms holds for Hermitian modular forms.

In this paper we show an index theorem for gerbes

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between…

We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozy on…

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between…

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

We prove a version of Szemeredi's regularity lemma for subsets of a typical random set in F_p^n. As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed.

We establish a version of the Beurling-Pollard theorem for operator synthesis and apply it to derive some results on linear operator equations and to prove a Beurling-Pollard type theorem for Varopoulos tensor algebras. Additionally we…

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small prime p. We give a number of examples of…

We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.

We prove a variant of the abstract probabilistic version of Szemer\'edi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random…

Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to F_p, for infinitely many primes p, while preserving finitely many algebraic incidences of S. In this note we show that the converse…

In this paper, we shall consider the notion of hyperbolic semi norm which on a module $X$ to set of all positive hyperbolic numbers. We shall prove the characterization of continuity of hyperbolic semi norm in this setup. We shall prove…

We prove a Torelli-like theorem for higher-dimensional function fields, from the point of view of "almost-abelian" anabelian geometry.

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…

Let $q$ be a power of a prime and $\mathbb{F}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in $\mathbb{F}_q^2$. We also use our results on…

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The…

We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.