Discretized sum-product type problems: Energy variants and Applications
Abstract
In this paper, we provide estimates for the additive discretized energy of \[Σc∈ C |\(a1, a2, b1, b2)∈ A2× B2: |(a1 +cb1) - (a2 + cb2)| δ\|δ,\] 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.
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