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.
Each breakthrough opened a new tool for proving numbers transcendental. Lindemann proved π is transcendental in 1882, ending the squaring-the-circle problem.
Liouville's constant L = 0.110001000000000000000001... has 1s at positions 1!, 2!, 3!, 4!, ... and 0s elsewhere. Joseph Liouville constructed it in 1844 as the first explicit transcendental number, 29 years before Hermite proved e transcendental. His proof showed algebraic numbers cannot be approximated by rationals too accurately: the rapidly spreading 1s in L violate this bound. The construction elegantly demonstrated transcendentals exist without Cantor's later diagonal argument.