Intersection properties of typical compact sets

TL;DR

This paper investigates the geometric and arithmetic properties of typical compact sets, showing they lack certain patterns, intersect low-dimensional planes minimally, and exploring their hitting probabilities and arithmetic characteristics.

Contribution

It establishes new generic properties of compact sets regarding pattern avoidance, intersection limits, and probabilistic behaviors in Euclidean spaces.

Findings

Typical compact sets do not contain similar copies of given patterns.

A typical compact set in [0,1]^d intersects any (d-1)-dimensional plane in at most d points.

The paper analyzes hitting probabilities and arithmetic properties of typical compact sets.

Abstract

We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. We study the "hitting probabilities" of compact sets in the sense of Baire category. In the end we study the arithmetic properties of typical compact sets in $[0,1]$ and the "hitting probabilities" of continuous functions.