IfLet $a,b \in \mathbb{Z}$ are$a,b \in \mathbb{N}_{\ge 2}$ be two positive integers andthat are multiplicatively independent $(ab, N) = 1$(i.e., are not powers of the same integer). I have seen (Bourgain, Lindenstrauss, Michel, Venkatesh: Some effective results for $\times a\,\times b$. Ergodic Th. Dyn. Syst. 29(6) (2009) 1705--1722 [link to paper on Venkatesh's web page]) that if, denoting by $X$ is the set of $\times a \times b$$\times a \,\times b$ multiples of a fraction [1]: $$ X = \left\{ a^k \times b^l \times \frac{m}{N}: 0 < k, l < \log N \right\} $$$$ X = \left\{ a^k b^\ell \frac{m}{N}: 0 < k, \ell < \log N \right\} $$ these numbers are $\epsilon$$\varepsilon$-dense in the reals: $\displaystyle d = \min_{x \in X} |x - a| < \epsilon $$\displaystyle d = \min_{x \in X} |x - a| < \varepsilon $ with $$ \epsilon = \kappa \;(\log \log \log N)^{-\kappa'}$$$$ \varepsilon = \kappa \;(\log \log \log N)^{-\kappa'}.$$ This does not seem terribly assuring as this number is tending to zero very very slowly. If Given that $\log N \approx \# \text{digits} (N)$ then, we've got $$\log \log \log N = \log \log \big[ \# \text{digits}(N) \big]$$$$\log \log \log N = \log \log \big[ \# \text{digits}(N) \big].$$
Has this result improved much? Do we know anything about the constants $\kappa, \kappa'$ ?
