Balanced two-colorings of finite sets in the square I

Abstract

LetT(N) be the least integer such that one can assign ± 1’s to anyN points in the unit square so that the sum of these values in any rectangle with sides parallel to those of the square have absolute value at mostT(N). G. Tusnádi asked what could be said about the order of magnitude ofT(N). We prove

$$\log N \ll T(N) \ll (\log N)^4 .$$

In contrast, ifT*(N) denotes the corresponding quantity where rectangles of any possible orientation are considered, we have

$$N^{{1 \mathord{\left/ {\vphantom {1 {4 - \varepsilon }}} \right. \kern-\nulldelimiterspace} {4 - \varepsilon }}} \ll T^* (N) \ll N^{{1 \mathord{\left/ {\vphantom {1 {2 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {2 + \varepsilon }}} $$

for anyε > 0.