A large plane set intersecting lines in infinitely many directions in at most one point

Eiderman, Vladimir; Larsen, Michael

doi:10.1090/proc/15341

A large plane set intersecting lines in infinitely many directions in at most one point
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by Vladimir Eiderman and Michael Larsen

Proc. Amer. Math. Soc. 149 (2021), 1091-1098

DOI: https://doi.org/10.1090/proc/15341

Published electronically: January 22, 2021

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Abstract:

We prove that for every at most countable family $\{f_k(x)\}$ of real functions on $[0,1)$ there is a single-valued real function $F(x)$, $x\in [0,1)$, such that the Hausdorff dimension of the graph $\Gamma$ of $F(x)$ equals 2, and for every $C\in \mathbb {R}$ and every $k$, the intersection of $\Gamma$ with the graph of the function $f_k(x)+C$ consists of at most one point. We also construct a family of functions of cardinality continuum and a function $F$ with similar properties.

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