论文标题
对ERD \ H {O}的差异问题的反思
Reflections on the Erd\H {o}s Discrepancy Problem
论文作者
论文摘要
我们考虑一些与著名的ERD \ H {O}的差异问题有关的着色问题。一组表格$ a_ {s，k} = \ {s，2s，\ dots，ks \} $，带有$ s，k \ in \ mathbb {n} $，称为a \ emph {均值{均值{我们证明，对于每一个固定的$ k $，都存在$ 2 $颜色的$ \ mathbb n $，以便每个套装$ a_ {s，k} $ is \ emph {proballt balanced}（集合$ a_ {s，k} $中的红色和蓝色元素的数量在大多数情况下都不同）。这促使对ERD \ H {O}的各种限制版本的问题进行了反思，该问题是通过对参数$ s，k $的各种限制而获得的。在稍微不同的方向上，我们讨论了问题的\ emph {多数}变体，其中每个设置$ a_ {s，k} $的颜色应与集合中的第一个元素不同。这个问题出乎意料地导致了一些有关$ \ {+1，-1 \} $的完全乘法功能的深入问题。特别是，是否存在从上方界定的部分总和的功能。
We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every fixed $k$ there exists a $2$-coloring of $\mathbb N$ such that every set $A_{s,k}$ is \emph{perfectly balanced} (the numbers of red and blue elements in the set $A_{s,k}$ differ by at most one). This prompts reflection on various restricted versions of Erd\H {o}s' problem, obtained by imposing diverse confinements on parameters $s,k$. In a slightly different direction, we discuss a \emph{majority} variant of the problem, in which each set $A_{s,k}$ should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in $\{+1,-1\}$. In particular, whether there is such a function with partial sums bounded from above.
