Comments on Tao's March 2001 Notices article
This page is addressed mostly to those mathematicians who have been
watching the controversy regarding Tao's article and are not quite
sure what it is about. So here is what I came up with after
going through just the first three pages - two if you don't count
the pictures. The rest of the article is not much different,
I just have not had the time to write a detailed report on it.
The main problem is that the article (a) does not tell the whole
story, and (b) does not say so. Also the facts are not always correct.
A related issue is that a lot of people were either not given
due credit at all or credited only for very small parts of what they
have actually done. This includes Stein, Wolff, Christ, Tomas, Oberlin,
Knapp, Cordoba, Seeger, Mockenhaupt, Schlag, Sogge, Carbery, Sjolin, and
many many others. It would have been impossible to acknowledge
all contributions in just a few pages, but why not at least say that
such work exists. And some contributors should have been mentioned anyway
- I would particularly like to know why Stein's name never appears
in the restriction/Bochner-Riesz part of the article, even though Stein
basically fathered and grandfathered that whole area of research.
"Page x, line y/z" will mean line z in column y on page x.
- 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
"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
- 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 N6/7 for the Erdos distance problem,
obtained by Solymosi and Toth.) None of them are mentioned.