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). 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.

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). 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.

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). Well before either Harman's 1997 book and the 2003 paper referenced by 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.

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). 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.