Comments on Tao's March 2001 Notices article

• Let's start with the title: "Emerging Connections between Combinatorics, Analysis, and PDE". Actually, connections between combinatorics and analysis have been around for decades and are not limited to the areas of research discussed in the article. (See e.g. Alex Iosevich's article in the June issue of the Notices.) Not to mention the long-standing close relationship between analysis and PDE. Of those connections that the article describes, the only ones that are actually emerging right now are those between additive number theory and Kakeya sets. They are interesting enough not to need questionable advertising.
  
  
• page 294, line 1/1: "In 1917 Kakeya posed the Kakeya needle problem (...) In 1927 the problem was solved by A. Besicovitch ..." See also page 298, line 2/4: "Historically, the first applications of the Kakeya problem to analysis arose in the study of Fourier summation in the 1970s." Really? Besicovitch himself tells a very different story in Amer. Math. Monthly 70 (1963), 697-706. Here is how K.J. Falconer summarizes it in his book The geometry of fractal sets:
  
  "In 1917 Besicovitch was working on problems on Riemann integration, and was confronted with the following question (...) noticed that if he could construct a compact set F of plane Lebesgue measure zero containing a line segment in every direction, this would lead to a counter-example (...) Besicovitch (1919) succeeded in constructing a set (...) with the required properties (...) the construction was later republished in Mathematische Zeitschrift (1928).
  
  "At about the same time, Kakeya (1917) and Fujiwara-Kakeya (1917) mentioned the problem of finding the area of the smallest convex set (...) Shortly after Besicovitch's departure from Russia in 1924, it was realized that a simple modification to the Besicovitch set yielded a solution (...) and the problem was solved in an unexpected manner".
  
  Thus: (1) Besicovitch constructed his "Kakeya set" in 1919, not 1927, and in fact published a paper about it in a Russian journal in 1920; (2) he did not even know about Kakeya's question at the time; (3) he then solved Kakeya's problem in 1925 and published his solution in 1928; (4) applications of Besicovitch sets to analysis are as old as Besicovitch's construction itself.
  
  For further pre-1970 appearances of Kakeya sets in analysis, see e.g. Busemann and Feller, Fund. Math. 22 (1934), 226-256. See also Stein and Weiss, Trans. AMS 140 (1969), 34-54, where a closely related construction of Nikodym (1927) is used to disprove the unrestricted convergence of Poisson integrals.
  
  
• page 294, line 2/15: "not Besicovitch's original construction" is actually Perron's "tree" construction (1928).
  
  
• page 295, line 1/12: "It is known that the delta-neighbourhood of a Besicovitch set in R² must have area..." Also page 295, line 1/-4: "It remains open in three and higher dimensions..." The Kakeya conjecture (Hausdorff version) in dimension 2 was proved by Davies (1971), who used an argument based on Marstrand's projection theorem. The geometrical argument alluded to later on is due to Cordoba (1977).
  
  
• page 295, line 19: "Recall that a bounded set E has Minkowski dimension..." In fact this is the definition of the upper Minkowski dimension. There is also a lower Minkowski dimension, and the two need not be equal.
  
  
• page 295, line 1/-12: "(There is also a corresponding conjecture for the Hausdorff dimension, but for simplicity we shall not discuss this variant.)" Actually why not? Most mathematicians know what Hausdorff dimension is (which cannot be said of the Minkowski dimension). Also the Hausdorff problem is more difficult, and arguably more important, than Minkowski. It would not have been technical or complicated to state the best Hausdorff bounds as well. There is moreover a third formulation of the conjecture, in terms of maximal functions. This one might indeed be too technical for an expository article; see however below. For the record, the best Hausdorff bounds at the time the article was written were (n+2)/2 in low dimensions (Wolff 1994) and (6n+5)/11 in high dimensions (Katz-Tao 1999). The best maximal function bounds at the time were Wolff's (n+2)/2 in low dimensions and Bourgain's (1/2+r)n for some r>0 (not computed explicitly in the paper) in high dimensions.
  
  
• page 295, line 2/1: the (4n+3)/7 part is due to Katz-Tao (1999), as explained on the following pages. The (n+2)/2+10^-10 part is due to Katz, Tao, and myself (1999) for n=3, and to Tao and myself (2000) for n>3.
  
  
• page 296, line 1/8: "it is fairly straightforward to show that the Minkowski dimension of Besicovitch sets is at least (n+1)/2 ..." The geometrical argument mentioned here is due to Bourgain. Other (n+1)/2 arguments (via x-ray transforms) arose earlier in the work of Drury, Christ, Duoandikoetxea, Rubio de Francia. In fact their x-ray estimates are much stronger than just bounds on the Minkowski dimension of Kakeya sets. I'm not sure exactly where the (n+1)/2 exponent came up first - if you happen to know that, please let me know!
  
  
• page 296, line 2/17: "For instance, the lower bound of (n+2)/2 for the Minkowski dimension was shown in 1995 by Wolff..." It was shown in 1994, and published (after the usual delay) in 1995. More importantly, Wolff's result applied not only to the Minkowski dimension (as this sentence seems to suggest) but also to the Hausdorff and maximal function problems. This is a significant distinction, see the comments for page 295, line 1/-12 and page 296, line 2/15. Wolff's Hausdorff and maximal function results are still the best in low dimensions.
  
  
• page 296, line 1/-15: "However, there appears to be a limit..." See also page 296, line 1/-4: "but they are clearly insufficient to resolve the full conjecture", and page 296, line 2/-4: "it does not seem sufficient to solve the problem". Cordoba's argument (mentioned earlier) is clearly insufficient to solve the problem in any dimension other than 2. Yet it appears throughout almost all of the subsequent literature on Kakeya and x-ray problems, and will not go away any time soon. Similarly, Wolff's "hairbrush" argument may be insufficient by itself, but it is a very basic argument which has since become an integral part of how we think about the problem, and therefore I consider it as a great achievement. If the geometrical methods have their limitations, then so does everything else, and I see no point in emphasizing this repeatedly.
  
  Just for fun, I will bring up an example that should be well known to the people in this field. After Roth proved Szemeredi's theorem for arithmetic progressions of length 3 using the "circle method", it did not seem possible that such methods would extend to the general case. That is, until Gowers did extend them. This led not only to quantitative improvements, but also (less directly) to the recent Kakeya developments. The point is, you never know.
  
  
• page 296, line 1/-11: "More sophisticated geometric analysis..." This is due to Katz, Tao, and myself (1999) for n=3, and to Tao and myself (2000) for n>3; many of the underlying geometrical arguments were due to Tom Wolff (the 1997 x-ray paper as well as personal communication).
  
  
• page 296, line 2/15: "replacing the quantity... (the relevant conjecture here is known as the Kakeya maximal function conjecture)". This gets mentioned somewhere between beta-sets, finite geometry analogues, and distance sets. So let's try to straighten it out.
  
  The maximal function conjecture is an important part of the "Kakeya mainstream" - more specifically, it is this particular formulation of the Kakeya conjecture that leads to progress on the restriction and Bochner-Riesz problems, see page 299, line 2/-4. It is not just one of "a much larger family of problems of a similar flavour". There is also a class of problems which are directly related to Kakeya sets and to the analysis questions discussed later, but never get mentioned in the article (unless this is what the author means by "light rays" in the same paragraph), namely x-ray and k-plane transform estimates. This line of research goes back to the work of Marstrand, Falconer, and Oberlin-Stein; see also remarks for page 296, line 1/8. The most recent results are due to Wolff, Tao and myself, Erdogan and Christ.
  
  Distance sets and "Kakeya-type" problems for circles are important in their own right, not as "variants" of Kakeya proper. It is indeed not clear how any of these problems - except of course for the maximal functions - are related to Kakeya. There are some heuristic arguments that suggest such connections; however, they "do not adapt well to the continuous Kakeya setting because of the difficulty in discretizing both addition and multiplication simultaneously". (This is from page 298, line 6.) Tom Wolff was aware of this issue and in fact mentioned it in his Prospects in Mathematics article. On the other hand, the "Kakeya for circles" problems are known to be relevant to some of the PDE questions discussed later in the article (such as local smoothing).
  
  And by the way, there are plenty of people who worked on these problems, including Marstrand, Bourgain, Wolff, Falconer, Mattila, Mitsis, Szekely, Sjolin, and many others. (The most recent development is probably an improved bound N^6/7 for the Erdos distance problem, obtained by Solymosi and Toth.) None of them are mentioned.