On the binary digits of the Erdős-Borwein constant

Abstract

In a landmark paper on arithmetical properties of Lambert series, Erdős proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erdős-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 Erdős 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.