Tower-type bounds for Roth's theorem with popular differences

一个热门多维非线性Roth定理中的界

Bounds in a popular multidimensional nonlinear Roth theorem

本研究得到非线性Roth定理的多维版本，为相关多维多项式Szemerédi定理情形的有效化迈出第一步，还证明了其有效热门版本，且该结果的量化依赖为指数型，异于线性情形的塔型界。

Tower tableaux and Schubert polynomials

We prove that the well-known condition of being a balanced labeling can be characterized in terms of the sliding algorithm on tower diagrams. The characterization involves a generalization of authors' Rothification algorithm. Using the characterization, we obtain descriptions of Schubert polynomials and Stanley symmetric functions.

Popular progression differences in vector spaces

Green proved an arithmetic analogue of Szemerédi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $ε$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $Θ(\log(1/ε))$…

Popular progression differences in vector spaces II

Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $α>0$, $β<α^3$, and prime number $p$, there is a least positive integer $n_p(α,β)$ such that if $n \geq n_p(α,β)$, then for every subset of $\mathbb{F}_p^n$ of density at least $α$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $β$. We determine for $p \geq 19$ the tower height of $n_p(α,β)$ up to an absolute constant factor and an additive term depending only on…

再探HJ数

HJ numbers revisited

研究哈雷 - 朱厄特数，核心方法是改进基于字母表大小归纳步骤为2 - 指数运算，主要贡献是将哈雷 - 朱厄特数边界改进为指数塔形式，还涉及相关定理及密度版本等。

On The Discrepancy of Quasi-progressions

The 2-colouring discrepancy of arithmetic progressions is a well-known problem in combinatorial discrepancy theory. In 1964, Roth proved that if each integer from 0 to N is coloured red or blue, there is some arithmetic progression in which the number of reds and the number of blues differ by at least (1/20) N^{1/4}. In 1996, Matousek and Spencer showed that this estimate is sharp up to a constant. The analogous question for homogeneous arithmetic progressions (i.e., the ones containing 0) was raised by Erdos in the 1930s, and it is still not known whether the discrepancy is unbounded. However…