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arXiv · 2604.02984

A Bilinear Kakeya Inequality in the Heisenberg Group

Yannis Galanos
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Abstract

We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in $\R^2$. By adapting an argument of F\"assler, Pinamonti and Wald involving Heisenberg projections, we show that the latter implies the former. We prove the estimate for curved tubes using a combination of techniques developed by Pramanik, Yang and Zahl, Wolff and Schlag. We introduce a novel broadness hypothesis inspired by works of Zahl, which rules out bush-type configurations that break transversal structure. We argue that such a hypothesis is needed for proving the bilinear estimates we present. We also introduce necessary additional linear terms to the estimate to counteract Szemer\'edi--Trotter-type clustering phenomena.

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Yannis Galanos. 2026-04-03. A Bilinear Kakeya Inequality in the Heisenberg Group. https://arxiv.org/abs/2604.02984

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