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.)

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