I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small \delta = \log(3) / \log(2) $ for this. We know, that the convergents of that cf correspond with good approximations, and that at high entries $\small c_k $ in the vector of the cf-coefficients that approximations are specifically well.
Now I observe, that the log $\small \gamma_k = \log(c_k) $ seems to have a linear relation with the following criterion.
Using that $\small S=\lceil N \cdot \delta \rceil $ we have
$\qquad \small t_N=\log_2(q_N) = \lceil N \cdot \delta \rceil - N\cdot \delta = 1- \lbrace N \cdot \delta \rbrace$
where the braces indicate the fractional part of a number. (Small t show good approximations of $\small 2^S $ to $\small 3^N $).
Moreover, the negative of the logarithms $\small \tau_N = \log(t_N) $ at the interesting(!) small t correspond roughly with N, so I define $\small r_N = \log(N)+ \log(t_N) $ as my approximation criterion, and "high" negative $\small r_N $ indicate good relative approximations ( I get best $\small r_N \sim -10.99 $ where N is a 10000-digit number with $\small \log_{10}(N)\sim 10853.91593 $ by the 21151'th coeffcient in the continued fraction with the value $\small c_{21151}= 59599 $).

If I select a couple of $\small c_k \gt 1000 $ compute $\small N_k $ and $\small r_{N_k} $ and do a linear regression between $\small \log(c_k) $ as x-vector and $\small r_{N_k} $ as y-vector I get a very well fitting linear equation of about $\small \hat y_k = -0.999782 \log (c_k) -0.002232443 $ where the absolute devitations from the empirical values are smaller than $\small 0.0002 $

That linear relation is so near a perfect fit, that I suspect, the error might be due to approximation or some minor additional coefficients. So my question is: do I have a -possibly trivial- relation between the coefficients of the continued fraction and the given criterion? Or is this just "by chance"? (And if it is a somehow trivial relation which I missed: what is its exact formula?)

After looking at $\small \hat c_k $ (see updated table below) and the observation, that $\small \hat c_k $ is in between $\small (1+ c_k) \pm 1 $ I'm beginning to seriously suspect, that the coefficients $\small c_k$ and the $\small \exp(-r_{N_k}) $ must indeed be related by the continued fraction mechanism, and the error is possibly simply determined by the residual after the truncation at the k'th convergent... but I still don't have it exactly.

For reference: I used Pari/GP, the continued fraction based on decimal precision of 50000 digits; below the table with some $\small c_k>1000 $ and the related $\small r_{N_k}$ which were used for linear regression (the x and y-column were used) $\small \begin{array} {rr|ll|l} k & c_k & x_k=\log(c_k) & y_k=r_{N_k} & (\log_{10}(N))&\hat c_k=\exp(- r_{N_k})&\epsilon_k=\hat c_k-c_k\\ \hline \\ 1605 &1192&7.083387848&-7.084328355 &&1193.121612&1.121&\\ 1677&1924&7.562161631&-7.562714985 &&1925.064947&1.064&\\ 3873&1943&7.571988449&-7.57276718 &&1944.513664&1.513&\\ 331&2436&7.798112629&-7.798451689 & 167.4206435 &2436.826092&0.826&\\ 3393&3095&8.037543185&-8.037852289 &&3095.956826&0.956&\\ 529&3308&8.104099056&-8.104621485 & 272.3639732&3309.728647&1.728&\\ 2765&4878&8.492490579&-8.492782912 & 1407.086282&4879.42621&1.426&\\ 4313&8228&9.015298251&-9.015346029 & 2234.175312&8228.39313&0.393&\\ 48539&23824&10.07844875&-10.07847168 &&23824.54623&0.546&\\ 21151&59599&10.99539407&-10.9954086 & 10853.91593 &59599.8659&0.865&\\ \end{array} $

  • $\begingroup$ Is $c_k$ the $k$th term of the continued fraction expansion of $\log(3)/\log(2)$? $\endgroup$ – MJD Mar 28 '12 at 13:18
  • $\begingroup$ @markDominus: yes $\endgroup$ – Gottfried Helms Mar 28 '12 at 13:31
  • $\begingroup$ Look at Kintchin's book on continued fraction, or the Pell equation, after which it may seem more plausible and perhaps also even provable. $\endgroup$ – bgins Mar 28 '12 at 14:55

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