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Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We provethat, if the irrationality exponent of $\xi$ is equal to $2$ or slightlygreater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple',in a suitable sense. Our result applies, among other classical numbers, tobadly approximable numbers, non-zero rational powers of ${\mathrm e}$, and$\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large.It establishes an unexpected connection between the irrationality exponent of areal number and its $b$-ary expansion.

On the $b$-ary expansions of $\log (1 + \frac{1}{a})$ and ${\mathrm e}$