相关论文: Furstenberg--S\'{a}rk\"{o}zy theorem and partition…

We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^\alpha(1-x)^{\beta} P - Q$, has at most $\deg P +\deg Q + 2$ roots in the interval $]0,~1[$. As a consequence, we…

We develop and study a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic…

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let…

Let $F\in\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a…

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of…

We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that…

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A| then A is…

We introduce the theory $\mathrm{PF}^{+,\times}$ of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the…

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4,…

In this note, we consider Szemer\'{e}di's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a…

We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of…

For a field $\mathbb{F}$ and integers $d, k$ and $\ell$, a set $A \subseteq \mathbb{F}^d$ is called $(k,\ell)$-nearly orthogonal if all vectors in $A$ are non-self-orthogonal and every $k+1$ vectors in $A$ contain $\ell + 1$ pairwise…

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field…

We settle the Polynomial Freiman--Ruzsa (PFR/Marton) conjecture for the integers and for cyclic groups. More precisely, we show that if $A$ is a finite subset of $\mathbb{Z}$ or $\mathbb{Z}/N\mathbb{Z}$ with $|A+A| \le K|A|$, then there is…

We show that if $E \subset \mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, and $|E| \geq \rho q^d$, where $ q^{-\frac{1}{2}}\ll \rho \leq 1$, then $E$ contains an isometric copy of at least $c…

The study of Property (B) starts as a special case of Lehmer's conjecture. An algebraic field is said to satisfy Property (B) if there exists a positive constant bounding by below the height of every point of infinite order. In this paper…

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting…