Fourier Analysis and Hausdorff Dimension

Mattila deserves kudos for having written an excellent text for the community of graduate students and research mathematicians with an analytic bent, one that exposes in considerable detail a particularly rich seam of mathematics at the interface between harmonic analysis and geometric measure theory in Euclidean space. The book focuses on various manifestations of the symbiotic relationship between the Fourier transform and Hausdorff dimension, and the reader will no doubt find an abundance of beautiful material to explore within this carefully crafted monograph.

Mattila’s earlier volume in the same series, Geometry of Sets and Measures in Euclidean Spaces (Cambridge, 1995), with close to 1300 citations on MathSciNet at present, has established itself as one of the standard references for geometric measure theory on the market. Mattila is a master expositor, and it is to be expected that his latest monograph will reap similar success as his earlier text.

To quote from the introduction:

In this book there are two main themes. Firstly, the Fourier transform is a powerful tool on geometric problems concerning Hausdorff dimension, and we shall give many applications. Secondly, some basic problems of modern Fourier analysis, in particular those concerning restriction, are related to geometric measure theoretic Kakeya (or Besicovitch) type problems.

The book is divided into four parts; I will provide a both highly incomplete and skewed overview. Part I, Preliminaries and some simpler applications of the Fourier transform, comprising approximately a quarter of the text, provides basic results on Borel measures in \(\mathbb{R}^n\) and the Fourier transform. It goes on to describe the basic connections between the latter and the Hausdorff dimension, e.g. the fact that the Hausdorff dimension of a Borel subset \(A \subset \mathbb{R}^n\) may be computed as the supremum of the numbers \(s\) such that there exists a Borel measure \(\mu\) (with compact support contained in \(A\), and with \(0 < \mu(A) < \infty\)) for which \[ \int |x|^{s-n} |\widehat{\mu}(x)|^2 \mathrm{d}x < \infty . \] There are useful chapters on Slices of measures and intersections with planes and Intersections of general sets and measures. Other highlights include a discussion of Marstrand’s projection theorem and Falconer’s distance set problem.

Part II, Specific constructions, which takes up close to a fifth of the text, begins with a proof of the beautiful result that the Fourier transform of the natural measure on a symmetric Cantor set with dissection ratio \(d\) goes to zero at infinity if and only if \(1/d\) is not a Pisot number. For readers intrigued by this result, the classic references remain eminently readable: Jean-Pierre Kahane and Raphael Salem’s Ensembles parfaits et séries trigonométriques, as well as Salem’s later Algebraic numbers and Fourier analysis, see Rudin’s 1964 BAMS review.

Other highlights in Part II are the proofs due to Peres-Simon-Solomyak and Kenyon of the fact that almost all orthogonal projections (onto lines) of a four-corner planar Cantor set have length zero, as well as part of Solomyak’s landmark result regarding Bernoulli convolutions. Regarding the latter, see Peres-Schalg-Solomyak’s masterful turn-of-the-century survey Sixty years of Bernoulli convolutions, as well as Varjú’s 2016 Recent progress on Bernoulli convolutions that goes beyond the chronological scope of Mattila’s text. Besicovitch and Nikodym sets are constructed, and the reader is introduced to the infamous Kakeya conjecture and the (less widely known) Nikodym conjecture, regarding which I will say more below. There is also a neat construction of Salem sets (sets that have equal Fourier and Hausdorff dimensions) via results on the almost sure decay of Fourier transforms of certain measures on trajectories of Brownian motion.

Part III, Deeper applications of the Fourier transform, returns the reader to a more sophisticated study of Fourier transforms with a view to strengthening results on geometric problems involving Hausdorff dimension that were discussed earlier, e.g. exposing parts of Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions (Duke Math. J. 102 (2000), no. 2, 193-251) and proving the best known dimension results for the distance set problem. Parts of the material in Part III (e.g., the proof of Wolff-Erdoğan estimates on the decay of certain spherical averages) will probably challenge the neophyte lacking experience with arguments from harmonic analysis.

The reader interested in moving to connections between harmonic analysis and the Kakeya conjecture can skip these chapters in the third movement and move directly to the denouement in Part IV, Fourier restriction and Kakeya type problems. This final part comprises close to two-fifths of the text proper, and is perhaps where the pulse of living mathematics lies closest to the book’s surface. Here the reader will find a thorough discussion of the incredible network of conjectures that surround Stein’s restriction problem and the Kakeya conjecture. The restriction problem began with Stein’s seminal work from the 1960s that uncovered the surprising fact that one is able to restrict the Fourier transform of \(L^p\) functions to certain measure zero sets such as certain curved hypersurfaces like the sphere, but not to hyperplanes. The statement of the restriction conjecture is somewhat technical, and I refer the interested reader to the articles linked to below. The Kakeya conjecture, which is far easier to state, asserts that measure zero subsets of \(\mathbb{R}^n\) (with \(n>2\)) that contain a unit line segment in every direction (such sets are called Besicovitch sets) must have full Hausdorff dimension. Note that the existence of Besicovitch sets is non-trivial, though it is a fact that can be established via various means today (see Section 11.6, “Further Comments”, for an excellent survey). As mentioned above, Mattila does provide a neat construction of such sets (following Falconer).

As an introduction to some of the issues involved, aimed at a non-specialist reader, I suggest sampling Terence Tao’s From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE (Notices AMS 48 (2001), no. 3, 294-303) coupled with Izabella Łaba’s insightful Comments on Tao’s March 2001 Notices article. Those interested in the extent of the subject’s reach to additive combinatorics should study Łaba’s survey From harmonic analysis to arithmetic combinatorics (Bull. AMS (N.S.) 45 (2008), no. 1, 77-115).

One should mention that there are, sadly, no exercises. So instructors planning on using the text to teach a graduate course will have to create their own. I also felt that the text would benefit from the inclusion of more diagrams, though this is more a reflection of my own approach to teaching mathematics than a criticism of Mattila’s style. Each of the 25 chapters in the book ends with an excellent Further Comments section, which provide food for thought and pointers to articles. Such sections would be especially useful if one were using the text to lead a seminar.

Suffice it to say that my brief review does little justice to the wide scope and excellent exposition that are hallmarks of Mattila’s style. Libraries should be encouraged to buy their copies in haste.

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.