$L^p$ averages of the discrete Fourier transform and applications

Abstract

The discrete Fourier transform has proven to be an essential tool in many geometric and combinatorial problems in vector spaces over finite fields. In general, sets with good uniform bounds for the Fourier transform appear more `random' and are easier to analyze. However, there is a trade-off: in many cases, obtaining good uniform bounds is not possible, even in situations where many points satisfy strong pointwise bounds. To address this limitation, the first named author proposed an approach where one attempts to replace the need for uniform ($L^\infty$) bounds with suitable bounds for the $L^p$ average of the Fourier transform. In subsequent joint work, the authors applied this approach successfully to improve known results in Fourier restriction and the study of orthogonal projections. In this survey we discuss this general approach, give several examples, and exhibit some of the recent applications.

Figures (6)

• Figure 1: Line through $(0,0)$ generated by $(3,1)$ in $\mathbb{F}_{23}^2$.
• Figure 2: Circle of radius $2$ centered at $(0,0)$ in $\mathbb{F}_{23}^2$.
• Figure 3: Parabola in $\mathbb{F}_{23}^2$.
• Figure 4: The threshold for which $S_0^3$ forms a $(p,s)$-Salem set is $s = \frac{1}{3} + \frac{1}{3p}$ (see Theorem \ref{['sphere_zero_salem']} with $n=4$), and this is plotted as a solid line. It is asymptotic to $\frac{1}{3}$ as $p \to \infty$. The trivial lower bound $(p, \frac{1}{p})$ from Corollary \ref{['p,1/p corolalry']} is plotted as a dashed line for comparison. One can quickly see that the sphere of radius zero exhibits good Fourier behavior on average, despite not being Salem.
• Figure 5: The threshold at which $\mathbb{F}_q^2 \setminus (\mathbb{F}_q \times \{0\})$ forms a $(p,s)$-Salem set is $s = \frac{1}{2} + \frac{1}{2p}$ (see Theorem \ref{['smallcomplement']} with $n=2$ and $k=1$), and this is plotted as a solid line. It is asymptotic to $\frac{1}{2}$ as $p \to \infty$. The trivial lower bound $(p, \frac{1}{p})$ from Corollary \ref{['p,1/p corolalry']} is plotted as a dashed line for comparison.

Theorems & Definitions (44)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Corollary 3.4
• Lemma 4.1
• Proposition 4.2