2
$\begingroup$

Let $a,b \in \mathbb{N}_{\ge 2}$ be two integers that are multiplicatively independent (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, denoting by $X$ the set of $\times a \,\times b$ multiples of a fraction: $$ X = \left\{ a^k b^\ell \frac{m}{N}: 0 < k, \ell < \log N \right\} $$ these numbers are $\varepsilon$-dense in the reals: $\displaystyle d = \min_{x \in X} |x - a| < \varepsilon $ with $$ \varepsilon = \kappa \;(\log \log \log N)^{-\kappa'}.$$ This does not seem terribly assuring as this number is tending to zero very very slowly. Given that $\log N \approx \# \text{digits} (N)$, 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 $\kappa, \kappa'$ ?

Improve this question
$\endgroup$
2
  • $\begingroup$ The constants $\kappa,\kappa'$ are very explicit, and can be deduced from the computation done in the paper (they are computed explicitly from some constants appearing in Baker's inequality). The bound itself have not been improved (to the best of my knowledge), nevertheless, if the order of $<a,b>$ is large, the famous exponential sum estimate of Bourgain-Konyagin-Gilbichuk gives you much better rates. People believe the later is the correct one, but the iterated log estimates in the BLMV paper follows from Baker's, which is essentially sharp, so definitely some new ingredients are needed. $\endgroup$
    – Asaf
    Commented Jun 15, 2018 at 18:42
  • $\begingroup$ Bourgain (in his EMS conference paper) mentions that studying the order of $<2,3>$ mod $p$ is a challenging problem. $\endgroup$
    – Asaf
    Commented Jun 15, 2018 at 18:42

0

Reset to default

You must log in to answer this question.