论文标题
算术进展和不规则离散结构的偏差概率
Deviation probabilities for arithmetic progressions and irregular discrete structures
论文作者
论文摘要
令随机变量$ x \，：= \，e（\ Mathcal {h} [b]）$计数由随机$ m $ m $ - element子集$ b $ to hypergraph $ \ mathcal {h} $的边缘数量。集中于$ \ Mathcal {h} $中的顶点程度差异很大的情况，我们证明了$ x $远非其平均值的概率。可以将这些结果应用于离散结构，例如$ \ {1，\ dots，n \} $中的$ k $ term算术进度。此外，我们的主要定理允许我们通过独立于概率$ p $将每个顶点纳入每个顶点来推断出$ b \ sim b_p $的结果。在此设置中，我们对算术进行的结果扩展了Bhattacharya，Ganguly，Shao和Zhao \ Cite {BGSZ}的结果。我们还提到了与相关的中央限制定理的连接。
Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$-element subset $B$ of its vertex set. Focussing on the case that the degrees of vertices in $\mathcal{H}$ vary significantly we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the $\{1,\dots, N\}$. Furthermore, our main theorem allows us to deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$. In this setting our result on arithmetic progressions extends a result of Bhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connections to related central limit theorems.
