We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a @math -H "older continuous map @math , with @math . Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in @math or @math in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as H "older curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves @math . We then find conditions in terms of tube approximations that ensure the approximating curves converge to a H "older curve. As an application, we provide sufficient conditions that guarantee fractional rectifiability of pointwise doubling measures in @math .
We present a generalisation of C. Bishop and P. Jones' result in [BJ1], where they give a characterisation of the tangent points of a Jordan curve in terms of @math numbers. Instead of the @math Jones' @math numbers, we use an averaged version of them, firstly introduced by J. Azzam and R. Schul in [AS1]. A fundamental tool in the proof will be the Reifenberg parameterisation Theorem of G. David and T. Toro (see [DT1]). In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes).