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We prove that for all integers 2≤m≤d−1, there exists doubling measures on ℝd with full support that are m-rectifiable and purely (m−1)-unrectifiable in the sense of Federer (i.e. without assuming μ≪m). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: N axis-parallel squares of side length s pack inside of a square of side length ⌈N1/2⌉s. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each m∈{2,3,4} and s0, f(E) has Hausdorff dimension s, and μ(f(E))>0. This is striking, because m(f(E))=0 for every Lipschitz map f:E⊂ℝm→ℍ1 by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space 𝕏 of Assouad dimension strictly less than m is a Lipschitz image of a compact set E⊂[0,1]m. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions. 

abstract: We examine caloric measures $$\omega$$ on general domains in $$\RR^{n+1}=\RR^n\times\RR$$ (space $$\times$$ time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of Taylor and Watson (1985) that the lower parabolic Hausdorff dimension of $$\omega$$ is at least $$n$$ and $$\omega\ll\Haus^n$$. On the other hand, we prove that the upper parabolic Hausdorff dimension of $$\omega$$ is at most $$n+2-\beta_n$$, where $$\beta_n>0$$ depends only on $$n$$. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). Heuristically, we show that the \emph{density} of obstacles in a cube needed to make it unlikely that a Brownian motion started outside of the cube exits a domain near the center of the cube must be chosen according to the ambient dimension. In the course of the proof, we give a caloric measure analogue of Bourgain's alternative: for any constants $$0<\epsilon\ll_n \delta<1/2$ and closed set $$E\subset\RR^{n+1}$$, either (i) $$E\cap Q$$ has relatively large caloric measure in $$Q\setminus E$$ for every pole in $$F$$ or (ii) $$E\cap Q_*$$ has relatively small $$\rho$$-dimensional parabolic Hausdorff content for every $$n<\rho\leq n+2$$, where $$Q$$ is a cube, $$F$$ is a subcube of $$Q$$ aligned at the center of the top time-face, and $$Q_*$$ is a subcube of $$Q$$ that is close to, but separated backwards-in-time from $$F$$: \begin{gather*} Q=(-1/2,1/2)^n\times (-1,0),\quad F=[-1/2+\delta,1/2-\delta]^n\times[-\epsilon^2,0),\\[2pt] \text{and }Q_*=[-1/2+\delta,1/2-\delta]^n\times[-3\epsilon^2,-2\epsilon^2]. \end{gather*} Further, we supply a version of the strong Markov property for caloric measures. 

Abstract For every$$n\geq 2$$, Bourgain’s constant$$b_n$$is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most$$n-b_n$$for every domain in$$\mathbb {R}^n$$on which harmonic measure is defined. Jones and Wolff (1988,Acta Mathematica161, 131–144) proved that$$b_2=1$$. When$$n\geq 3$$, Bourgain (1987,Inventiones Mathematicae87, 477–483) proved that$$b_n>0$$and Wolff (1995,Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing$$b_n<1$$. Refining Bourgain’s original outline, we prove that$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$for all$$n\geq 3$$, where$$c>0$$is a constant that is independent ofn. We further estimate$$b_3\geq 1\times 10^{-15}$$and$$b_4\geq 2\times 10^{-26}$$. 

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^2 (P. Jones, 1990), in R^n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' beta numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^n that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space. 

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s . At the endpoint, α = s , a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/ s )-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s -dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n , the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n .