棋盘上一根针的差异，II
The discrepancy of a needle on a checkerboard, II
摘要：
把平面看作一个棋盘，每个单位正方形以任意方式涂上黑色或白色。在以前的论文中，我们证明了，对于任何这样的着色有直线段，任意大的长度，使他们的白色长度减去他们的黑色长度的差异，在绝对值，至少是他们的长度的平方根，直到一个乘法常数。对于相应的"有限"问题（$N \times N$棋盘），我们已经证明，我们可以用这样一种方式来着色，即对于线段的任何位置，上述数量至多为$C \sqrt{N \log N}$。在这个后续行动中，我们表明，它是可能的颜色的无限棋盘用两种颜色，使任何线段$I$的一种颜色超过另一个是由$C_ε\Abs{I}^{\frac12+ε}$以上的界限，任何$ε>0$。我们还证明了下界的圆弧的差异。最后，我们做了一些观察，关于$L^p$的差异段和弧，$p<2$，我们的$L^2$为基础的方法无法给出任何合理的估计。
英文摘要：
Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding "finite" problem ($N \times N$ checkerboard) we had proved that we can color it in such a way that the above quantity is at most $C \sqrt{N \log N}$, for any placement of the line segment. In this followup we show that it is possible to color the infinite checkerboard with two colors so that for any line segment $I$ the excess of one color over another is bounded above by $C_ε\Abs{I}^{\frac12+ε}$, for any $ε>0$. We also prove lower bounds for the discrepancy of circular arcs. Finally, we make some observations regarding the $L^p$ discrepancies for segments and arcs, $p<2$, for which our $L^2$-based methods fail to give any reasonable estimates.
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。
