Measurements of interdependence in fractional Cartesian products

The notion of a fractional Cartesian product and a subsequent notion of combinatorial dimension appeared first in a harmonic-analytic context in the course of filling ”analytic” gaps between successive (ordinary) Cartesian products of spectral sets [1, 2]. (Detailed accounts of this, and much more, appear in [4].) Combinatorial dimension is an index of interdependence: attached to a subset of an ordinary Cartesian product, it marks precisely the interdependence of restrictions to the set, of the canonical projections from the Cartesian product onto its independent coordinates. We can gauge, analogously, also the interdependence of restrictions to the same set, of projections from the Cartesian product onto interdependent coordinates of a prescribed fractional Cartesian product. We thus obtain distinct indices of interdependence associated, respectively, with distinct fractional Cartesian products. In this article we survey results, found in [5], concerning relations between these indices. We recall some definitions. Let E1, ..., En be sets, and F ⊂ E1 × · · · × En. (We refer to E1 × · · · × En as the ambient product of F .) For integers s > 0, define ΨF (s) = max{| F (A1 × · · · ×An) |: Ai ⊂ Ei, | Ai |≤ s}, (1)