Are there infinitely many $n$ such that the decimal expansion $2^n$ begins with $n$?


For example, $2^6=64$ and $2^{10} =1024$.


It can easily be shown that this problem is equivalent to the following.


Are there infinitely many $n$ such that 
$\{n\log\_{10}2-\log\_{10}n\}<\log\_{10}{(1+\frac{1}{n})}$? 


Here, $\{x\}=x-[x]$, where [x] denotes the largest integral number that is smaller than $x$.


More generally, for $\alpha,\beta,\gamma > 0$, are there infinitely many $n$ such that 
$\{\alpha n - \beta \log n\}<\frac{\gamma}{n}$?


I think it would help if I know something about the distribution of $\{\log n\}$.



$n$ such that decimal digits of $2^n$ begins with $n$
Some general information on the problem, which is probably open.


As pointed out by Ivan Neretin at math.stackexchange, this is [OEIS A100129](http://oeis.org/A100129). There you can find the first 16 numbers with this property.


According to this paper,


* Jan van de Lune, "[A note on a problem of Erdős](http://oai.cwi.nl/oai/asset/7417/7417A.pdf)" (1978)


the problem goes back to Erdős:



> 
> "Recently my attention was drawn to the following observation made by
>  P. Erdős: $2^6=64$ and $2^{10}=1024$. Here we have two examples of the
>  phenomenon that the number $2^n$ starts with the same ordered sequence
>  of digits as the natural number $n$ itself."
> 
> 
> 


The characterization used in van de Lune's calculations is


$$0\leq \{n\log 2\}-\{\log n\}<\log(n+1)-\log n$$


