Lines in Every Direction with No ee-Random Points

TL;DR

This paper proves that in every direction in the Euclidean plane, there exists a line containing no ee-random points, showing the existence of lines with points that are algorithmically predictable with double exponential time resources.

Contribution

It effectivizes the construction of lineal extensions of Kakeya sets to show lines with no ee-random points, resolving an open question by Lutz and Lutz (2015).

Findings

Existence of lines with no ee-random points in every direction.

Lines contain points with predictable locations for double exponential time algorithms.

Addresses an open problem in algorithmic randomness and geometric measure theory.

Abstract

We prove that in every direction in the Euclidean plane, there exists a line containing no double exponential time random (ee-random) points. This means each point on these lines has an algorithmically predictable location, to the extent that a gambler in an environment with fair payouts can, using double exponential time computing resources, amass unbounded capital placing bets on increasingly precise estimates of the point's location. Our proof relies on effectivizing the construction of the lineal extension of a Kakeya set. This resolves an open question of Lutz and Lutz (2015).