Two-point patterns determined by curves

Harmonic Analysis/Fractals Hybrid Seminar

Two-point patterns determined by curves

Abstract: Given a smooth function $\Phi \colon \mathbb{R} \rightarrow \mathbb{R}^d$, we ask whether every Borel subset of $\mathbb{R}^d$ with sufficiently high Hausdorff dimension contains a pair of the form $\{x, x+\Phi(t)\}$ with $\Phi(t) \neq 0$. We prove that a large class of ``curved” functions have this property while certain ``flat” functions do not. This is joint work with Malabika Pramanik.

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