Let $\scr E$ denote the set of irrational numbers whose continued fraction convergents $p_j/q_j$ obey the rule $$\log q_{j+1} \le (\log q_j)^{1 + o(1)} \quad (j \to \infty).$$ Thus $\scr E$ contains the algebraic numbers and its complement has Hausdorff dimension 0. Denoting by $\|\cdots\|$ the distance to the nearest integer, the authors obtain $$\min_{d|n} \|d\theta\|= \frac 1{\tau(n)^{1+o(1)}} \quad \text{a.e.},$$ that is, for a set of natural numbers $n$ of density 1. The proof involves the estimation of exponential sums $$\sum_{n\le x} z^{\Omega(n)}e(n\theta)$$ and an excursion into the theory of modified Dirichlet $L$-functions of the form $$L(s, \chi; y) = \sum_{P^+(n) \le y} \frac{\chi(n)}{n^s},$$ where $P^+(n)$ denotes the greatest prime factor of $n$. For example, the authors prove that $$\sum\Sb \chi \pmod q \\\chi \ne \chi_0\endSb |L(1, \chi; y)|^b \ll \phi(q) (\log\log 2q)^b$$ for any $b \ge 1$. (R.C. Baker)