关于拟级数的离散性
On The Discrepancy of Quasi-progressions
摘要：
等差数列的2-着色不等式是组合不等式理论中的一个著名问题。在1964年，罗斯证明了，如果从0到N的每个整数都是红色或蓝色，则存在一些等差数列，其中红色的数量和蓝色的数量至少相差（1/20）N^{1/4}。在1996年，Matousek和Spencer证明了这个估计值是一个常数。齐次等差数列的类似问题（即，包含0的那些）是由鄂尔多斯在20世纪30年代提出的，并且仍然不知道该差异是否是无界的。然而，很容易构造密度任意接近1的偏着色，使得所有齐次等差数列都有有界的差异。一个相关的问题涉及拟级数的差异。准级数由一个实数的连续倍数组成，每个倍数向下舍入到最接近的整数。在1986年，贝克表明，任何2-着色，准级数对应于几乎所有的实数在（1，\infty）有差异至少log* N，反塔功能。我们将其下界改进为（log N）^{1/4 - o（1）}，并证明了存在偏差至少为（1/50）N^{1/6}的拟级数.即使用正密度的部分着色代替2-着色，我们的结果仍然有效。
英文摘要：
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, it is easy to construct partial colourings with density arbitrarily close to 1 such that all homogeneous arithmetic progressions have bounded discrepancy. A related problem concerns the discrepancy of quasi-progressions. A quasi-progression consists of successive multiples of a real number, with each multiple rounded down to the nearest integer. In 1986, Beck showed that given any 2-colouring, the quasi-progressions corresponding to almost all real numbers in (1, \infty) have discrepancy at least log* N, the inverse of the tower function. We improve the lower bound to (log N)^{1/4 - o(1)}, and also show that there is some quasi-progression with discrepancy at least (1/50) N^{1/6}. Our results remain valid even if the 2-colouring is replaced by a partial colouring of positive density.
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