A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

Abstract

The study of the geometry of n-uniform measures in R d has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski-Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension n − 3.In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most n − 3.A measure Φ is n-rectifiable if it is absolutely continuous to H n and there exists a countable collection of C 1 n-manifolds {M j } j such that Φ(R d \ j M j ) = 0. In [P], Preiss proved the following remarkable theorem relating the rectifiability of a measure to its density.Theorem 1.1 ([P]). A Radon measure Φ of R d is n-rectifiable if and only if it satisfies the following property:The density Θ n (x) = lim r→0 Φ(B(x, r)) ω n r n exists and is positive andTo prove this theorem, Preiss studies the geometry of n-uniform measures which appear as tangents (blow-ups) to measures satisfying (1.2). A measure µ is said to be n-uniform if there exists a constant c > 0 such that for any x in the support of µ and any radius r > 0, we have:( 1.3) In [P], Preiss also provides a classification of the cases n = 1, 2 in R d for any d. In these cases, µ is n-Hausdorff measure restricted to a line or a plane respectively. Interestingly, flat measures are not the only examples of uniform measures. Indeed, in [KoP], Kowalski and Preiss proved that µ is (d − 1)-uniform in R d if and only if µ = H d−1 V where V is a (d − 1)-plane, or d ≥ 4 and there exists an orthonormal system of coordinates in which µ = H d−1 (C × W ) where W is a (d − 4)-plane and C is the KP-cone. The classification for n ≥ 3 and codimension d − n ≥ 2 remains an open question.Kirchheim and Preiss later proved in [KiP] that the support Σ of a uniformly distributed measure (of which n-uniform measures are an example) is a real analytic variety, namely the intersection of countably many zero sets of analytic functions. An application of the stratification theorem for real analytic varieties implies that Σ must be a countable union of real analytic manifolds and the singular set has Hausdorff dimension at most (n − 1).We investigate the Hausdorff dimension of the singular set S µ of an n-uniform measure µ. Our main result is the following theorem.Theorem 1.2. Let µ be an n-uniform measure in R d , 3 ≤ n ≤ d. Then dim H (S µ ) ≤ n − 3.In the cases n = 3, d = 3, it is a standard result that the only 3-uniform measure (up to normalization) is 3-Lebesgue measure. In this case, the bound is obvious. To see that this bound cannot be improved, let n ≥ 3, d > n and consider the measure µ defined as:where M is the set M = (x 1 , . . . , x d ); x 2 4 = x 2 1 + x 2 2 + x 2 3 and x n+1 = . ...

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

“…For an in-depth introduction, we refer the reader to the exposition of Preiss' theorem by De Lellis [17]. The classi cation of m-uniform measures in R n is as of yet incomplete, but some progress has recently been made by Tolsa [59] and Nimer [47,48]. The deep connections between the existence of densities and recti ability of sets and absolutely continuous measures in Euclidean space, as described above, have been explored in metric spaces beyond R n by several authors; see e.g.…”

Rectifiability; a survey

“…Recently Nimer [Nim18] produced many interesting examples. See also [Tol15a], [Nim17] and [Nim19] for other results.…”

On uniform measures in the Heisenberg group

“…The first example is due to Kowalski and Preiss [14], who showed that the measure H 3 Σ is a 3-flat measure in R 4 , where Σ = {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 : x 2 1 = x 2 2 + x 2 3 + x 2 4 } is the light cone. In fact, Kowalski and Preiss have completely classified (n − 1)-uniform measures in R n for all values of n: every such measure is either (n−1)-flat or is proportional to H n−1 M where M is an (n − 1)-dimensional algebraic variety in R n which is isometric to Σ × R n n − 1 remains unknown, although recent work of Nimer [24], [25], [26] has improved our understanding and provided new examples. Kirchheim and Preiss [13] proved that the support of a uniform measure in R n is an analytic variety, and Tolsa [29] showed that such supports satisfy the David-Semmes 'weak constant density' condition and hence are uniformly rectifiable.…”