Related papers: Quasipolynomial bounds for the corners theorem

Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-coloring of {1,2...,n} contains a monochromatic (a,b)-triple.…

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS…

In this paper, we describe the relationship between the quasi-component q(G) of a (perfectly) minimal pseudocompact abelian group G and the quasi-component q(\widetilde G) of its completion. Specifically, we characterize the pairs (C,A) of…

Let $A=E \times E_{ss}$ be a principally polarized almost ordinary split abelian surface over a finite field $\mathbb{F}_{q}$. We give asymptotic upper and lower bounds on the number of principally polarized abelian surfaces over…

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem…

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each…

Let $G=\langle x^d+c_1,\dots,x^d+c_s\rangle$ be a semigroup generated under composition for some $c_1,\dots,c_s\in\mathbb{Z}$ and some $d\geq2$. Then we prove that, outside of an exceptional one-parameter family, $G$ contains a large and…

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a $\Delta$-coloring on…

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field,…

We define nondeterministic communication complexity in the model of communication complexity with help of Babai, Hayes and Kimmel. We use it to prove logarithmic lower bounds on the NOF communication complexity of explicit graph functions,…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…

In this paper we show some multiplicity estimates theorems for a connected algebraic group (not necessarily commutative) $G$ over an algebraically closed subfield of $\mathbb{C}$. More specifically, under particular assumptions on the…

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, minimized over all distributions on…

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if…

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\phi_1$ and $\phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with…

We explore multi-round quantum memoryless communication protocols. These are restricted version of multi-round quantum communication protocols. The "memoryless" term means that players forget history from previous rounds, and their behavior…