From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: Worms Date: 18 May 2000 16:56:39 -0400 Newsgroups: sci.math Summary: [missing] Dan Goodman [sci.math Thu, 18 May 2000 00:59:52 +0100] wrote > I've been looking at the following problem, which I've been told > is a hard research problem. I was wondering if anyone can tell > me where the current research has got to or provide me with some > links if there any. The problem is to find a set of minimum > measure such that any rectifiable (2d) curve of length 1 can be > translated, rotated, reflected so that it lies within the set. > Apparently, you can construct a set of zero measure that will > accomodate any polygonal curve, but someone has proved that the > set must have strictly positive measure for all rectifiable > curves. Any idea how these were proven? Thanks, > > Dan Goodman > > p.s. the subject is "worms" because the original problem was > stated with the word worms instead of rectifiable curves of > length 1. See Chapter 7 ("Besicovitch and Kakeya sets") of Falconer [5], especially 7.4: "Generalizations" on pp. 106-109. For instance, Theorem 7.9 on page 103 says that any subset of R^2 containing a line in every direction must have Hausdorff dimension 2. The result is due to Davies [4], who proved the slightly stronger result that any subset of R^2 containing a line segment in every direction (there does not have to be a positive lower bound on the lengths of these segments) must have Hausdorff dimension 2. [Indeed, as Falconer points out, if dim(E) < 2, then E intersects all lines, in all but a 1-measure zero set of directions, in a set of 1-measure zero.] I believe it is still an unsolved problem, the Kakeya Conjecture, as to whether there exists a set in R^n (n > 2) with n-measure zero containing a line segment in every direction. [See Bourgain [2], Green [11], and Wolff [9] [17] for some recent work on this question.] In the opposite direction Besicovitch (1919) constructed a compact set in R^2 with 2-measure zero containing a line segment in every direction. Various similar results have been proved over the years. For example, Davies [4] constructed a set in R^2 of 2-measure zero containing a translate of every polygonal arc, and Marstrand [7] constructed in each R^n a set of Hausdorff dimension 1 containing every congruent copy of any preassigned countable collection of lines. [Keep in mind that there are continuum many such congruent copies when n > 1.] Marstrand [8] proved that given any n-measure zero set E in R^n (n > 1), there exists a C^infinity curve of length one having no congruent copy in E. Note that any C^infinity curve--indeed, any C^1 curve--is rectifiable. [In fact, Marstrand proves that every non-empty open connected subset D of R^n contains a C^infinity curve which cannot be transformed into a subset of E by an invertible analytic mapping from D into R^n.] Falconer [5] (bottom of page 107) mentions this as solving the 'worm problem', although Marstrand [8] doesn't appear to use this term. It seems that the term 'worm problem' is due to Leo Moser. See Finch [10] for more on the worm problem. Finch gives 36 references, many of which are not among those I give below. A helpful beginning expository paper is Cunningham [3]. Thomas Wolff (CalTech) has recently written several papers on these matters. Of these papers, I would recommend looking at Wolff [9] [17], which contains 58 references. Other papers that might be of interest are Bourgain [2] and Green [11]. For the Baire category analog of a Nikodym set, see Bagemihl and Humke [1], Humke [6], and Steprans [15]. [1] Frederick Bagemihl and Paul Humke, "Rectifiably ambiguous points of planar sets", J. Australian Math. Soc. (A) 20) (1975), 85-109. [MR 51 #14004; Zbl 305.04005] [2] J. Bourgain, "On the dimension of Kakeya sets and related maximal inequalities", Geom. Funct. Anal. 9 (1999), 256-282. [MR 2000b:42013; Zbl 930.43005] [3] F. Cunningham, "Three Kakeya problems", Amer. Math. Monthly 81 (1974), 582-592. [MR 50 #14497; Zbl 286.52009] [4] R. O. Davies, "Some remarks on the Kakeya problem", Proc. Cambridge Phil. Soc. 69 (1971), 417-421. [MR 42 #7869; Zbl 209.26602] [5] Kenneth J. Falconer, THE GEOMETRY OF FRACTAL SETS, Cambridge Tracts in Mathematics 85, Cambridge University Press, 1985. [MR 88d:28001; Zbl 587.28004] [6] Paul Humke, "Baire category and disjoint rectilinear accessibility", J. London Math. Soc. (2) 14 (1976), 245-248. [MR 56 #563; Zbl 342.26005] [7] J. M. Marstrand, "An application of topological transformation groups to the Kakeya problem", Bull. London Math. Soc. 4 (1972), 191-195. [MR 47 #4234; Zbl 248.28017] [8] J. M. Marstrand, "Packing smooth curves in R^q", Mathematika 26 (1979), 1-12. [MR 81d:52009; Zbl 403.28008] [9] Thomas H. Wolff, "Recent work connected with the Kakeya problem", pp. 129-162 in Hugo Rossi (ed.), Prospects in Mathematics (Invited talks on the occasion of the 250'th anniversary of Princeton University, 1996.), American Math. Society, 1999. (This paper is on-line. See [15].) [MR 2000d:42010. This was a featured review, which even non-math-sci-doc members can access at ; The Zbl review not officially numbered yet, but you can find it on-line by going to .] |---------------------| | INTERNET REFERENCES | |---------------------| [10] Steven Finch, (a) "Moser's Worm Constant", (b) "Results for Closed Worms", and (c) "Outcomes Corresponding to Translation Covers". (a) (b) (c) [11] B. J. Green, "The Kakeya Problem", 78 page expository paper. [12] Nets Hawk Katz, Izabella Laba, and Terence Tao, "An improved bound on the Minkowski dimension of Besicovitch sets in R^3", 65 pages, to appear in Annals of Math. [13] Nets Hawk Katz and Terence Tao, "A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture", 6 pages, submitted to Math Research Letters. [14] Izabella Laba, "An improved bound on the Minkowski dimension of Besicovitch sets in R^3" and "An improved bound on the Minkowski dimension of Besicovitch sets in medium dimension". [15] Juris Steprans, "A category analogue of a Nikodym set", on-line construction in the link labeled "A partial solution to the problem" at [16] Terence Tao, (a) "Bochner-Riesz, Restriction, and Kakeya estimates" and (b) "Besicovitch sets" [A java applet giving one version of Besicovitch's construction.], (c) A very detailed chart showing the relationships between results and conjectures related to Kakeya and Nikodym sets [See the link for "interconnections between various conjectures in this family".], (d) "Restriction theorems and applications", Lecture Notes for UCLA Math 254B, Spring 1999. (a,c) (b) (d) [17] Thomas H. Wolff, "Recent work connected with the Kakeya problem" and "Lecture Notes for CalTech Math 191d-Topics in Harmonic Analysis" Spring 2000. Dave L. Renfro P.S. On a personal note, you might be interested to know that shortly after I began looking up these references--around 12 noon Central time--the Des Moines (Iowa) emergency sirens came on. I had been hearing distant thunder and was thinking of getting off-line and unplugging my computer if it got worse but, until the sirens came on, I was not aware that a severe storm and tornado was headed my way. I immediately turned on my radio and heard that a tornado had been sighted less than 15 miles from where I live! Immediately ... Computer is shut down, computer is unplugged from the wall, anything I considered to be especially valuable that was near any of my apartment windows (expensive math books, birth certificate and SS card, etc.) was moved to the hallway, etc. The radio reported strong hail, producing white-out conditions downtown downtown, less than 2 miles from where I live, but I never saw any hail. From what I could judge from the radio reports the tornado came within about 12 or 13 miles of where I live. However, at no time did I see any significant wind. In fact, most of the time it was simply very dark with light rain and almost no wind. After an hour or so the situation appeared sufficiently safe for me to get back on-line. I'm not sure what amount of damage has occurred in the areas east to me because I've been absorbed with Kakeya stuff since then (!). [This subject is somewhat related to things I'm interested in. Because it's been about a year since I last looked into what people are doing in this area, responding to this post provided me with an excuse to see what has been going on lately.]