棋盘上一根针的差异
The discrepancy of a needle on a checkerboard
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。我们证明，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），我们也证明了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$C \sqrt{N \log N}$。
英文摘要：
Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show 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 also prove 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.
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。
