Afternoon workshop on Fourier analysis, fractals, and finite fields

Abstract: Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over $\mathbb{F}_q$, the finite field with $q$ elements. For a subset $E \subseteq \mathbb{F}_q^d$, the distance set of $E$ is defined as $\Delta(E) := \{\lVert x - y \rVert : x, y \in E\}$, where the “norm” $\lVert \cdot \rVert : \mathbb{F}_q^d \to \mathbb{F}_q$ is given by $\lVert \alpha \rVert = \alpha_1^2 + \dots + \alpha_d^2$ for $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb{F}_q^d$.

Iosevich, Koh, and Parshall proved that if $d \geq 2$ is even and $|E| \geq 9q^{d/2}$, then $\mathbb{F}_q = \frac{\Delta(E)}{\Delta(E)}$, where $\frac{\Delta(E)}{\Delta(E)} = \left\{\frac{a}{b} : a \in \Delta(E),\ b \in \Delta(E) \setminus \{0\}\right\}$.

In this talk, we will discuss the geometric interpretation of this result, its natural generalizations, and other related questions.

Abstract: It is well-known that when sets have the same Fourier and Hausdorff dimensions there are no exceptions to Marstrand’s projection theorem. In this talk we will show how we can use the method of dimension interpolation to improve the state of the art estimates for the Hausdorff dimension of the exceptional set of projections in higher dimensions. Joint work with Jonathan Fraser.

Abstract: We talk about Bochner--Riesz operators associated with general convex domains in the plane. We see how bounds for these operators depend on the shape of the boundary of the domain. Domains that have flat boundary such as polygons yield stronger bounds compared to domains that have curved boundary like circles. For the Bochner--Riesz problem, this notion of flatness/curvedness is captured by the affine dimension of the domain, which we shall discuss in our talk. Another feature of the boundary that affects the boundedness of these operators, is its additive structure. Using ideas from additive combinatorics, we will construct domains with a Cantor-like boundary, that yield strong bounds for the associated Bochner--Riesz operators.

Abstract: Given two self-similar measures a,b on the real line, consider the multiplication map m: (x,y)->xy. It is interesting to consider the image measure m(a,b). In particular, it is curious to see when the image measure is a.c. with respect to Lebesgue. If the dimensions of a,b are too small, then by dimension consideration, the image measure cannot be a.c. On the other hand, if the dimensions are large, almost nothing is known. In this talk, I will survey some results in this direction including new progress in a recent joint work with Amlan Banaji.

Abstract: Stein's spherical maximal function theorem is a foundational result in harmonic analysis concerning the behaviour of averages of functions over spheres. The original proof relies on Fourier analysis in an essential way. In this talk, I will describe a cute argument which can prove results of this type and is purely based on geometric/combinatorial arguments (no Fourier analysis!).

Abstract: For many self-similar measures, if we ignore a sparse set of bad frequencies if necessary, then the Fourier transform of the measure will decay to 0 quite rapidly as the norm of the (remaining) frequencies grows. This property of decay outside sparse frequencies has a variety of important applications. One way to prove this property is to use an argument that originated in work of Erdos and Kahane. We will describe this argument and see that it is relatively straightforward to implement if the linear parts of the contractions in the iterated function system are all equal. If not all linear parts are equal, however, then to prove the desired property one can first disintegrate the measure into many other measures, most of which are amenable to an Erdos-Kahane argument. This talk is based on (separate) joint work with Simon Baker and Han Yu.

Abstract: The Fourier transform can be used to tackle various combinatorial problems cast in the setting of vector spaces over finite fields. This often has parallels with the Euclidean setting, where the combinatorial problem might take the form of a familiar geometric measure theory problem. I will introduce some of these ideas with emphasis on situations where working on one side of this parallel can help inform work on the other.