Ones in Liouville's constant: exponentially sparse
A 1 appears at positions 1, 2, 6, 24, 120, 720... (the factorials). All other positions are 0. The gaps grow exponentially: after position 24 the next 1 is at position 120.
Methods for proving transcendence: Liouville opened the door
Each breakthrough opened a new tool for proving numbers transcendental. Lindemann proved π is transcendental in 1882, ending the squaring-the-circle problem.