## What's new:

Feb 22, 2007

Feb 16, 2007

• Solved: The puzzle described in “Squares, cubes, and smooth functions” was already solved by Henri Joris (Arch. Math. (Basel) 39 (1982), no 3., 269-277).  Thanks to Catalin Bedea for the reference.  I have updated the article to reflect a proof of the puzzle inspired by that paper.

Feb 13, 2007

Feb 7, 2007

• Uploaded: “What is good mathematics?”, submitted, Bull. Amer. Math. Soc.  This (my first attempt at writing meta-mathematics!) is a solicited article on the (possibly dangerous!) topic of what constitutes good mathematics.  My answer is a bit complicated: there are many possible metrics to measure quality mathematics, but the relative importance of these metrics depends on the state and nature of the field, and evolves over time.  Also, one has to see how a piece of mathematics fits into the context of a larger mathematical story.  I illustrate these points with the story of Szemeredi’s theorem.
• Uploaded: The short story “Squares, cubes, and smooth functions”.  This is a description of a cute little problem I heard from Brian Conrad a while back: if f: R -> [0,+infty) is such that f^2 and f^3 is smooth, is f itself smooth?  I don’t know the answer, but it ought to be decidable.  I don’t foresee any major applications of this little puzzle, but it was amusing enough to be worth sharing.

Jan 26, 2007

• Congratulations to Tim Flannery for being selected as 2007 Australian of the Year.  (I had a lot of fun attending that event, even if a short and harmless conversation I had there managed to make it to the local paper :-).)

Jan 21, 2007

• Uploaded: “A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields”, with Ben Green, submitted, J. Aust. Math. Soc.  Here we improve the bounds on the Freiman and Balog-Szemeredi-Gowers theorems (which describe the behaviour of sets with small sumset, or sets with many additive quadruples) in the model setting of the finite field geometry F_2^n.  By using an iterative approach to gradually improve the doubling constant or additive energy, until the sets involved are “flat” enough for Fourier techniques to be effective, we obtain improved exponential (but still non-polynomial, unfortunately) bounds; more precisely, for sets A of doubling K, we obtain a large intersection with a subspace of cardinality K^{-O(sqrt(K))} |A|, and similarly for sets A with |A|^3/K additive quadruples we obtain a large intersection with a subspace of cardinality K^{-O(K)} |A|.   In a sequel to this paper we shall use these results to obtain an asymptotic for the best way to cover a set of small doubling in F_2^n by a subspace.

Jan 17, 2007

Jan 11, 2007

• Uploaded: The short story “Ornstein’s counterexample to an L^1 maximal inequality”.  An old maximal inequality of Stein asserts that the maximal operator sup_n P^n f is bounded on L^p for 1 < p <= infinity whenever P is a self-adjoint Markov operator.  It is natural to ask for the weak (1,1) analogue of this result, but an example of Ornstein shows that it fails.  Here we present Ornstein’s example, which is an interesting Markov process created by designing the percolation probabilities in order to “delay” the appearance certain portions of mass into a region until other portions have “diffused”.

Jan 9, 2007

• Uploaded: The short story “Wolff’s proof of the corona theorem”.  These are my notes on Wolff’s proof of the celebrated Corona theorem of Carleson, based on Gamelin’s exposition.  I replaced some of the complex-analytic arguments with real-analytic ones, and observed that by using Uchiyama’s constructive version of the Fefferman-Stein decomposition, the proof can in fact be made entirely constructive (e.g. no appeal to Hahn-Banach is necessary).  But in essential details, the argument is Wolff’s.