A Quantified Two-projection Theorem for Nonlinear Projections

Zhangze Li; Krystal Taylor

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arXiv · 2606.00381

A Quantified Two-projection Theorem for Nonlinear Projections

Zhangze Li·Krystal Taylor

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Abstract

The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. Further, we use a multiscale framework to obtain a quantitative version of our nonlinear two-projection theorem. Our arguments utilize methods introduced by Tao, who provided a quantitative treatment of the classic linear two-projection theorem.

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BibTeX RIS

Zhangze Li, Krystal Taylor. 2026-05-29. A Quantified Two-projection Theorem for Nonlinear Projections. https://arxiv.org/abs/2606.00381

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