On the binary digits of the Erd\H{o}s-Borwein constant

In a landmark paper on arithmetical properties of Lambert series, Erd\H{o}s proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erd\H{o}s-Borwein constant. Crandall, in 2012, studied properties of the base-2 expansion of this constant, and left the following as an open problem: Does the string $11$ occur infinitely often in the base-2 expansion of $E$? This open problem was also subsequently noted by Shallit. We succeed in introducing a full proof that solves Crandall's problem in the affirmative. Our proof combines a congruence construction in the spirit of Erd\H{o}s and an estimate due to Alford, Granville, and Pomerance for the counting function for primes in arithmetic progressions. Our argument was developed through extensive interactions with GPT-5.5 Pro.