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Wednesday 11 Oct 2023Dense Forests Constructed from Grids

Victor Shirandami - University of Manchester

Newman Purple 13:30-14:30

A dense forest is a subset $F$ of $\mathbb{R}^n$ with the property that for all $\epsilon > 0$ there exists a number $V(\epsilon) > 0$ such that all line segments of length $V(\epsilon)$ are $\epsilon$-close to a point in $F$. The function $V$ is called a visibility function of $F$. In the work of Adiceam, Solomon, and Weiss (2022) it was shown that, given an $\eta > 0$, there exists dense forests constructed from a finite union of grids which admit a visibility function of order $\epsilon^{-(n-1)-\eta}$. This is arbitrarily close to optimal in the sense that a finite union of grids admits only visibility functions bounded below at order $\epsilon^{-(n-1)}$.


In this talk we will first provide a necessary and sufficient condition for a finite union of grids to be a dense forest in terms of the irrationality properties of the matrices defining them. This result, however, does not provide an explicit visibility function. We provide a result to complement this, namely, given an $\eta > 0$ there exists a $k \in \mathbb{N}$ such that almost all unions of $k$ grids are dense forests admitting a visibility function of order $\epsilon^{-(n-1)-\eta}$. That is, the visibility achieved in the construction of Adiceam, Solomon, and Weiss almost always occurs. The notion of `almost all' is considered with respect to several underlying measures which are defined according to the Iwasawa decomposition of the matrices used to define the grids.


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