In his book *Metric Number Theory*, Glyn Harman mentions the following problem he attributes to Erdős:


Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that


$$\lim\_{N\rightarrow\infty} \frac{1}{\log N} \sum\_{n=1}^N \frac{1}{n}f(n\alpha) = \int\_0^1 f(x) dx$$


for almost all $\alpha$,


writing "so far as the author is aware, this question remains open."


Harman's book is from 1997. Does anyone know the current status of the problem?


**Motivation, for the curious**


We lose no generality in assuming $f$ has mean $0$. The rough idea is that for almost all $\alpha$, $n\alpha$ will be equidistributed $(\mod 1)$ in a strong enough way to cause a great deal of cancellation in the sum, so in particular we might guess the sum is $o(\log N)$. It is a weaker version of a more classical conjecture of Khintchine that 


$$\lim\_{N\rightarrow\infty} \frac{1}{N} \sum\_{n=1}^N f(n\alpha) = \int\_0^1 f(x) dx$$


for almost all $\alpha$, where $f$ is as above. This is known to be false. (Of course, if $f$ is continuous it is true, for all irrational $\alpha$ even.)



A question of Erdős on equidistribution
The statement was shown to be false by J. Bourgain in a paper published in 1988 (Almost Sure Convergence and Bounded Entropy, [doi:10.1007/BF02765022](http://dx.doi.org/10.1007/BF02765022)). Well before either Harman's 1997 book and the 2003 paper Gjergji mentioned in the comment, which both say that it is an open problem!


From page 2 of Bourgain's paper



> 
> As further application of our method, a problem due to A. Bellow and a question raised by P. Erdős are settled.
> 
> 
> 


And, further down (using ${\bf T}={\bf R}/{\bf Z}$),



> 
> The problem of Erdős mentioned above deals with weaker versions of the Khintchine problem. In particular he raised the question whether given a measurable subset $A$ of $\bf T$, then for almost all $x$ the set $\lbrace j\in{\bf Z }\_+\mid jx\in A\rbrace$ has a logarithmic density, i.e.
>  $$
> \frac{1}{\log n}\sum\_{\substack{j\le n\\\\ jx\in A}}\frac1j\to\vert A\vert.
> $$
>  We will disprove this fact.
> 
> 
> 


I can't vouch that Bourgain's paper is free of errors, as I have only just found it now and haven't read through it all in detail. However, Bourgain's result is also (very briefly) referred to in this 2004 paper, <http://arxiv.org/abs/math/0409001v1>, so I assume it is considered to be valid.


