Matching Random Subsets of the Cube with a Tight Control on One Coordinate

Abstract

Consider a measure $\mu$ on $\lbrack 0, 1\rbrack^2$, and $2n$ points $X_1, \cdots, X_n, Y_1, \cdots, Y_n$ that are independent and distributed according to $\mu$. Consider $2n$ points $U_1, \cdots, U_n, V_1, \cdots, V_n$ that are independent and uniformly distributed on $\lbrack 0, 1 \rbrack$. Then there exists a constant $K$ (independent of $\mu$) such that if $s \leq \sqrt n / K$, with probability close to 1 we can find a one-to-one map $\pi$ from $\{1, \cdots, n\}$ to itself such that $\forall i \leq n, \quad |U_i - V_{\pi(i)}| \leq \frac{K}{s},$ $\frac{1}{n} \sum_{i \leq n} |X_i - Y_{\pi(i)}| \leq K \big(\frac{s}{n}\big)^{1/2}.$