Additive equations over lattice points on spheres

Abstract: In this talk, we will consider additive properties of lattice points on spheres. Thus, defining $S_m$ to be the set of lattice points on the sphere $x^2 + y^2 + z^2 + w^2 = m$, we are interested in counting the number of solutions to the equation $a_1 + a_2 = a_3 + a_4,$ where $a_1, ..., a_4$ lie in some arbitrary subset $A$ of $S_m$. Such an inquiry is closely related to various problems in harmonic analysis and analytic number theory, including Bourgain's discrete restriction conjecture for spheres. We will survey some recent results in this direction, as well as describe some of the various techniques, arising from areas such as incidence geometry, analytic number theory and arithmetic combinatorics, that have been employed to tackle this type of problem.

number theory

Audience: researchers in the topic

Number theory by the sea

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