Discretized sum-product type problems: Energy variants and Applications

TL;DR

This paper extends sum-product problems to discretized energy estimates, providing bounds based on set non-concentration and employing Guth-Katz-Zahl methods, with applications discussed.

Contribution

It introduces new bounds for discretized energy in sum-product problems using Guth-Katz-Zahl techniques, with improved clarity and optimization.

Findings

Derived bounds depend on set non-concentration conditions

Extended sum-product estimates to discretized energy variants

Discussed multiple applications of the new estimates

Abstract

In this paper, we provide estimates for the additive discretized energy of \[\sum_{c\in C} |\{(a_1, a_2, b_1, b_2)\in A^2\times B^2: |(a_1 +cb_1) - (a_2 + cb_2)|\le \delta\}|_{\delta},\] that depend on non-concentration conditions of the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate changes along the way clarifying and optimizing many of the steps. Several applications will also be discussed.