Points on Computable Curves

• Authors: Xiaoyang Gu;Jack H. Lutz;Elvira Mayordomo
• Venue: FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2006

The "analyst's traveling salesman theorem" of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higher-dimensional Euclidean spaces by Okikiolu (1992), says that a bounded set K is contained in some curve of finite length if and only if a certain "square beta sum", involving the "width of K' in each element of an infinite system of overlapping "tiles" of descending size, is finite. In this paper we characterize those points of Euclidean space that lie on computable curves of finite length. We do this by formulating and proving a computable extension of the analyst's traveling salesman theorem. Our extension, the computable analyst's traveling salesman theorem, says that a point in Euclidean space lies on some computable curve of finite length if and only if it is "permitted" by some computable 'Jones constriction". A Jones constriction here is an explicit assignment of a rational cylinder to each of the abovementioned tiles in such a way that, when the radius of the cylinder corresponding to a tile is used in place of the "width of K" in each tile, the square beta sum is finite. A point is permitted by a Jones constriction if it is contained in the cylinder assigned to each tile containing the point. The main part of our proof is the construction of a computable curve of finite length traversing all the points permitted by a given Jones constriction. Our construction uses the main ideas of Jones's "farthest insertion" construction, but takes a very different form, because, having no direct access to the points permitted by the Jones constriction, our algorithm must work exclusively with the constriction itself.