Bounds for discrete maximal functions of codimension 3

Abstract: We study the bilinear discrete averaging operator $T_{\lambda}(f,g)(x) = \sum_{m,n \in V_{\lambda}} f(x-m) g(x-n)$, where $f$ and $g$ are functions in $\ell^p(\mathbb Z^d)$ and $\ell^q(\mathbb Z^d)$ and the summation is over the integer solutions $(m,n) \in \mathbb Z^{2d}$ of the equations \[ |m|^2 = |n|^2 = 2m \cdot n = \lambda, \] where $|\cdot|$ is the standard Euclidean norm on $\mathbb R^d$. We prove an approximation formula for the Fourier multiplier of $T_{\lambda}$ and establish the boundedness of the respective maximal operator from $\ell^p(\mathbb Z^d \times \ell^q(\mathbb Z^d)$ to $\ell^r(\mathbb Z^d)$ for a range of choices for $p,q,r$. Our work is related to classical work on simultaneous representations of integers by quadratic forms as well as to the study of point configurations in combinatorial geometry.

Joint work with T.C. Anderson and E.A. Palsson.

number theory

Audience: researchers in the topic

---

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

---