If $a_n$ and $a_n b_n$ are Cesàro summable 0-1 sequences, is $b_n$ Cesàro summable as well?

Question

My question is related to this one: Is the product of a Cesaro summable sequence of $0$s and $1$s Cesaro summable?

Let $(a_n)$ and $(b_n)$ be infinite sequences of zeros and ones. Assume that $\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n$ and $\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n b_n$ exist and are both strictly positive.

Can I conclude that $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n$$ exists?

Suppose $a_n = 0$ for all $n$. Do you have any information about $b_n$ then? — Daniel Fischer, Nov 14 '17 at 13:34
@DanielFischer you're right sorry, I added the assumption that the limits of the summations are strictly positive. — user52227, Nov 14 '17 at 13:52

score 2 · Accepted Answer · answered Nov 14 '17 at 14:02

Unless

$$\lim_{N\to \infty} \frac{1}{N}\sum_{n = 1}^N a_n = 1,$$

we cannot conclude that

$$\lim_{N\to \infty} \frac{1}{N}\sum_{n = 1}^N b_n$$

exists. If we let $C = \{ n : a_n = 0\}$, the conditions impose no restriction on the behaviour of $b_n$ on $C$. If $C$ has positive natural density, we can make the $b$-averages oscillate by having alternating long stretches of $b_n = 0$ and of $b_n = 1$ in $C$.

If $a_n$ and $a_n b_n$ are Cesàro summable 0-1 sequences, is $b_n$ Cesàro summable as well?

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