棋盘格测量的圆偏差
Circle discrepancy for checkerboard measures
摘要：
把这个平面看作是一个棋盘图案中全等单位正方形的联合，每个正方形以任意方式涂上黑色或白色。曲线与给定颜色的差异是其白色长度减去其黑色长度之差，以绝对值表示。我们证明了对于任何半径t>1，存在一个半径为t或2 t的整圆，对于某个数值常数c>0，偏差大于ct^（1/2）。我们还证明了对于任何t>1，存在一个半径恰好为t的圆弧，其偏差大于ct^（1/2）。最后，我们研究了更一般的曲线及其内部的相应问题。这些结果回答了Kolountzakis和Iosevich提出的问题。
英文摘要：
Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius t>1 there exists a full circle of radius either t or 2t with discrepancy greater than ct^(1/2) for some numerical constant c>0. We also show that for every t>1 there exists a circular arc of radius exactly t with discrepancy greater than ct^(1/2). Finally we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.
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。
