The Analyst's Traveling Salesman Theorem for large dimensional objects

The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a square sum of quantities called beta-numbers that measure how non-flat the set is at each scale and location. Conversely, given such a curve, the square sum of its beta-numbers is controlled by the total length of the curve, giving us quantitative information about how non-flat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional surfaces. This is joint work with Raanan Schul.

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