On logarithmically Benford sequences (Q2821722)

Let \({\mathbf a}=\{a_i\}_{i\in{\mathcal I}}\) be a sequence of nonzero real numbers indexed by an infinite subset \({\mathcal I}\subset{\mathbb N}\). If \(\sigma\) is a generalized asymptotic density, e.g. the standard asymptotic one \(d\), or the logarithmic one \(\delta\), then \(\sigma({\mathbf a},A)\), \(A\subset {\mathbb R}\) denotes the corresponding density of the subsequence \(\{a_i\in A\}\). The sequence \({\mathbf a}\) is \(\sigma\) Benford in base \(b\) if for any (nonzero) string of base \(b\) digits \(S_b\) (understood as an integer) we have \(\sigma({\mathbf a},b,S_b):=\sigma({\mathbf a},\left\{x\in{\mathbb R}:|x| \text{ begins with }S_b\) in base \(b\)