Analogues of the Lebesgue Density Theorem for Fractal Sets of Reals and Integers

Abstract

We prove the following analogues of the Lebesgue density theorem for two types of fractal subsets of U: cookie-cutter Cantor sets and the zero set of a Brownian path. Write C for the set, and jit for the positive finite Hausdorff measure on C. Then there exists a constant c (depending on the set C) such that for /x-almost every xeC,is the e-ball around x and d is the Hausdorff dimension of C. We also define analogues of Hausdorff dimension and Lebesgue density for subsets of the integers, and prove that a typical zero set of the simple random walk has dimension \ and density V(2/;r). a set of finite measure in a class of infinite ergodic measure-preserving transformations; this is joint work with J. Aaronson and M. Denker and will appear elsewhere. The theorem proved there is closely related to Chung and Erdos' work although the proof and interpretation are quite different.The purposes of the present paper are several. Firstly we introduce the notion of order-two density and develop its basic properties: consistency with respect to usual density (which however diverges almost surely for all the examples mentioned above); a Radon-Nikodym-like result for absolutely continuous measures; and the comparison with a hierarchy of order-n densities based on the Hardy-Riesz log averages (see [15]) and on the averaging operators of [14]. Secondly we introduce the techniques needed to prove existence of the order-two density for the examples mentioned above; the existence of order-2 density for the Hausdorff measure on these sets can be considered to be an analogue (for Hausdorff measure) of the Lebesgue density theorem. Finally, we want to show that there is a deep underlying connection between all of the techniques we use-even though they may at first seem disparate. The middle-third Cantor set is dealt with in some detail because it is possible there to show these connections. The analogies that one sees between the different situations are not precise but seem to be very helpful in suggesting problems and methods.The notion of order-two density is related to Mandelbrot's concept of lacunarity (see [24], especially pp. 315-318, for an intuitive description and illustrations). The lacunarity of a fractal should describe the degree to which the structure is fractured; one wants a way of comparing different sets of the same dimension or related sets of different dimensions. Order-two density provides a possible tool for making such comparisons. In the physics literature Smith, Fournier and Spiegel [33] observe that estimates of fractal dimension (they consider in particular the correlation dimension) can show log-oscillatory behaviour. When such oscillations occur, this brings added difficulties to the problem of numerical estimation of dimension. Smith, Fournier and Spiegel are suggesting that one can however make use of this oscillation as a way of measuring the 'textural property of fractal objects that Mandelbrot calls lacunarity'. But as they point out, if the sets are not strictly self-similar then in general the oscil...

Lacunarity of self-similar and stochastically self-similar sets

“…It turns out that lim →0 −(d−D) |K( )| does not exist in many interesting cases, like for example the usual ternary Cantor set (see section 4.1). However, the limit lim T →∞ T −1 T 0 e (d−D)t |K(e −t )|dt always exists for self-similar sets and one may use this quantity instead as a measure of lacunarity, if one is to summarize this information into a single number (compare with Bedford and Fisher [1]). We stress that our aim is to prove results on existence; the question of whether the proposed quantities are 'good measures of lacunarity' is a different issue, requiring separate investigation.…”

Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d

“…, x N } and s > 0, the Riesz s-energy of ω N is defined by (where δ x denotes the unit atomic measure at x) converges to µ s in the weak-star topology on M(A) as N → ∞. We use a starred arrow to denote weak-star convergence, that is, for s ∈ (0, d) we have (1) This research was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0808093. 1 In the case s ≥ d, the discrete minimal energy problem is well-posed even though the continuous problem is not.…”

“…We use a starred arrow to denote weak-star convergence, that is, for s ∈ (0, d) we have (1) This research was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0808093. 1 In the case s ≥ d, the discrete minimal energy problem is well-posed even though the continuous problem is not. Recently, asymptotic results for the discrete minimal energy problem were obtained in [5] and [2] …”

“…. ,K N ⊂ R d and a sequence of biLipschtiz mapsφ 1 We next find upper bounds for each of the terms in (13-16). First, Lemma 3.5 implies that, for s ∈ (s 1 , d), expression (13) is less than ε/N .…”

Second order ergodic theorems for ergodic transformations of infinite measure spaces

“…Examples where the log-averages converge have been given in [4,8] (compare also [7]). Chung and Erdös [5,Theorem 6] proved that the Chung-Erdös averages converge for any conservative, ergodic Markov shift.…”