On the $b$-ary expansion of a real number whose irrationality exponent is close to 2

Let $b\ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324\dots$, then the $b$-ary expansion of $\xi$ cannot be `too simple’, in a suitable sense. This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017].