If $a,b \in \mathbb{Z}$ are two positive integers and $(ab, N) = 1$ I have seen that if $X$ is the set of $\times a \times b$ multiples of a fraction [1] $$ X = \left\{ a^k \times b^l \times \frac{m}{N}: 0 < k, l < \log N \right\} $$ these numbers are $\epsilon$-dense in the reals: $\displaystyle d = \min_{x \in X} |x - a| < \epsilon $ with $$ \epsilon = \kappa \;(\log \log \log N)^{-\kappa'}$$ This does not seem terribly assuring as this number is tending to zero very very slowly. If $\log N \approx \# \text{digits} (N)$ then we've got $$\log \log \log N = \log \log \big[ \# \text{digits}(N) \big]$$

Has this result improved much? Do we know anything about the constants $\kappa, \kappa'$ ?