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.)

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.)

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

$$\lim_{N\rightarrow\infty} \frac{1}{\log N} \sum_{n=1}^N \frac{1}{n}f(n\alpha) = \int_0^1 f(\alpha) d\alpha$$

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

$$\lim_{N\rightarrow\infty} \frac{1}{N} \sum_{n=1}^N f(n\alpha) = \int_0^1 f(\alpha) d\alpha$$

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.)

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.)