Combinatorial dimension in fractional Cartesian products

The combinatorial dimension relative to an arbitrary fractional Cartesian product is defined. Relations between dimensions in certain archetypal instances are derived. Random sets with arbitrarily prescribed dimensions are produced; in particular, scales of combinatorial dimension are shown to be continuously and independently calibrated. A combinatorial concept of cylindricity is key. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 26, 146–159, 2005 1. DEFINITIONS, STATEMENT OF PROBLEM The idea of a fractional Cartesian product and a subsequent measurement 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 [2, 3]. (Detailed accounts of this, and much more, appear in [5].) Succinctly put, combinatorial dimension is an index of interdependence. Attached to a subset of an ordinary Cartesian product, it gauges precisely the interdependence of restrictions to the set, of the canonical projections from the Cartesian product onto its independent coordinates. We can analogously gauge 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 interpendences associated, respectively, with distinct fractional Cartesian products. A question naturally arises: What are the relationships between these various indices? Correspondence to: R. Blei © 2004 Wiley Periodicals, Inc. 146 COMBINATORIAL DIMENSION IN FRACTIONAL CARTESIAN PRODUCTS 147 To make matters precise, we first recall, and then extend basic notions found in Chapters XII and XIII of [5]. Let E1, . . . , En be sets, and let F ⊂ E1 × · · · × En. (We refer to E1 × · · · × En as the ambient product of F.) For integers s > 0 define F(s) = max{|F ∩ (A1 × · · · × An| : Ai ⊂ Ei, |Ai| ≤ s, i ∈ [n]}, (1) where [n] = {1, . . . , n}. For a > 0, define dF(a) = sup{ F(s)/s : s = 1, 2, . . .}. (2) The combinatorial dimension of F is dim F = sup{a : dF(a) = ∞} = inf{a : dF(a) < ∞}. (3) Next we define the fractional Cartesian products. For S ⊂ [n], let πS denote the canonical projection from E1 × · · · × En onto the product whose coordinates are indexed by S, πS( y) = ( yi : i ∈ S), y = ( y1, . . . , yn) ∈ E1 × · · · × En. Let U = (S1, . . . , Sm) be a cover of [n] (i.e., S1 ⊂ [n], . . . , Sm ⊂ [n], and ∪j=1 Sj = [n]), and define a fractional Cartesian products based on U to be (E1 × · · · × En)U = {(πS1(y), . . . , πSm(y)) : y ∈ E1 × · · · × En}. We view (E1 × · · · × En)U as a subset of E1 × · · · × Em , and measure its combinatorial dimension by solving a linear programming problem ([4, 8]): If E1, . . . , En, are infinite sets, and αU = max   n ∑ i=1 xi : xi ≥ 0, ∑ i∈Sj xi ≤ 1 for j ∈ [m]   , then dim(E1 × · · · × En)U = αU. (4) Examples 1. (Maximal and minimal fractional Cartesian products) For integers 1 ≤ k ≤ n, let U be a cover that is an enumeration of all k-subsets of [n], and let V be a cover of [n], all of whose elements are k-subsets of [n] such that, for every i ∈ [n], |{S ∈ V : i ∈ S}| = k. Then, αU = αV = n/k, and (taking E1 = · · · = En = N) we obtain from (4) dim(N)U = dim(N)V = n k . The ambient product of (N)U is (n k ) -dimensional, and the ambient product of (N)V is n-dimensional. Generally, if k and n are relatively prime, then the dimension of the ambient product of any nk -dimensional fractional Cartesian product is at least n, and no greater than (n k ) . 2. (Random constructions) Fractional Cartesian product are subsets of ambient products typically of high dimension. To wit, there are no nontrivial fractional Cartesian product in N, whereas for arbitrary α ∈ (1, 2) there exists an abundance of random sets F ⊂ N with dimF = α [6,7]. How to produce deterministically F ⊂ N with dimF = α, where α ∈ (1, 2) is arbitrary, is an open problem.