Matching Theorems, I

Abstract

In Chapter 4 we demonstrate that the generic chaining (or of course some equivalent form of it) is already required to really understand the irregularities occurring in the distribution of N points (X _i ) _i≤N independently and uniformly distributed in the unit square. These irregularities are measured by the “cost” of pairing (=matching) these points with N fixed points that are very uniformly spread, for two notions of cost. For each of them we prove that the expected cost of an optimal matching is bounded by $L (\log N)^{\alpha}/\sqrt{N}$ for either α=1/2 or α=3/4. The factor $1/\sqrt{N}$ is simply a scaling factor, but the fractional powers of log are indeed fascinating (and optimal). The connection with Chapter 2 is that one can often reduce the proof of a matching theorem to the proof of a suitable bound for a quantity of the type $\sup_{f\in \mathcal{F}} | \sum_{i\leq N}(f(X_{i}) -\int f{\rm d} \lambda)|$ where $\mathcal{F}$ is a class of functions on the unit square and λ is Lebesgue’s measure. That is, we have to bound a complicated random process. The main issue is to control in the appropriate sense the size of the class $\mathcal{F}$. For this we parametrize this class of functions by a suitable ellipsoid of Hilbert space using Fourier transforms. This approach illustrates particularly well the benefits of an abstract point of view: we are able to trace the mysterious fractional powers of log back to the geometry of ellipsoids in Hilbert space.