Lines in Every Direction with No ee-Random Points

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).