论文标题
单位立方体子集的szemerédi-type定理
A Szemerédi-type theorem for subsets of the unit cube
论文作者
论文摘要
我们调查了$ n $ term算术进度的差距$ x，x+y，\ ldots，x+（n-1）y $内部的积极度量子集$ a $ a $ a $ a $ a $ a $ a $ a $ [0,1]^d $。 If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $ d $，以及$ a $的措施。已知的反例阻止了该结果对指数$ p $的其余值的概括。我们还为上述间隔的长度提供了明确的界限。证明使界限取决于Szemerédi定理中整数的当前可用界限，该定理用作黑匣子。证明的关键要素是类似于多线性希尔伯特变换的操作员的功率类型取消估计值。作为该方法的副产品，我们获得了$ n $二维的侧面长度的相应（以前已知）结果的定量改进，顶点为$（[0,1]^2）^n $的阳性量表。
We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemerédi's theorem on the integers, which are used as a black box. A key ingredient of the proof are power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive measure subset of $([0,1]^2)^n$.
