A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. prove a version Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman Theorem this setting by showing any lower d-regular set \(E \subseteq H\) that $$\begin{aligned} \textrm{diam}(E)^d + \beta ^d(E) \sim \mathscr {H}^d(E) \text {Error}, \end{aligned}$$where \(\beta ^d(E)\) give measure the curvature E error term is related to theory uniform (a introduced David Semmes). To do this, we show how modify Reifenberg Parametrization Toro so it holds space. As corollary, uniformly rectifiable if only satisfies so-called Bilateral Weak Geometric Lemma, meaning bi-laterally well approximated planes at most scales locations.