On the b-ary expansions of log (1+1/a) and e

Abstract

Let b >= 2 be an integer and xi be an irrational real number. We prove that, if the irrationality exponent of xi is equal to 2 or slightly greater than 2, then the b-ary expansion of xi cannot be "too simple", in a suitable sense. Our result applies to, among other classical numbers, to badly approximable numbers, non-zero rational powers of e, and log(1 + 1/a), provided that the integer a is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its b-ary expansion.