arXiv math/07031082007-03-05math.NT

两个等差数列和的离散性

Discrepancy of Sums of two Arithmetic Progressions

Nils Hebbinghaus

备注：: 15 pages, 0 figures

摘要：

估计集合$[N]=\{1,2,\hdots,N\}$中所有等差数列的超图的差异度是组合差异理论中著名的公开问题之一。这个经典超图的一个推广是$k$（$k\geq 1$ fixed）等差数列和超图。这个超图的超边的形式为$[N]$中的$A_{1}+A_{2}+\hdots+A_{k}$，其中$A_{i}$是等差数列。Hebbinghaus（2004）证明了这个超图的一个下界$Ω(N^{k/(2k+2)})$。注意，概率方法对于所有固定的$k$给出阶数$O((N\log N)^{1/2})$的上界。2005年，PADIivativa将所有$k\geq 3$的下限提高到了$Ω(N^{1/2})$。因此，$k=2$（两个等差数列之和的超图）仍然是唯一一个已知上界和下界之间存在较大差距的情况。我们桥他的差距（对数因子）证明了一个下界的顺序$Ω(N^{1/2})$的差异的超图的两个等差数列的总和。

英文摘要：

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set $[N]=\{1,2,\hdots,N\}$ was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of $k$ ($k\geq 1$ fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form $A_{1}+A_{2}+\hdots+A_{k}$ in $[N]$, where the $A_{i}$ are arithmetic progressions. For this hypergraph Hebbinghaus (2004) proved a lower bound of $Ω(N^{k/(2k+2)})$. Note that the probabilistic method gives an upper bound of order $O((N\log N)^{1/2})$ for all fixed $k$. Přívětivý improved the lower bound for all $k\geq 3$ to $Ω(N^{1/2})$ in 2005. Thus, the case $k=2$ (hypergraph of sums of two arithmetic progressions) remained the only case with a large gap between the known upper and lower bound. We bridge his gap (up to a logarithmic factor) by proving a lower bound of order $Ω(N^{1/2})$ for the discrepancy of the hypergraph of sums of two arithmetic progressions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文