(exercise 1.3.10 of additive combinatorics)

Problem:

Prove that there does not exist $k, m \geq 1$, (where $k,m$ are not both 1) and $B \subset N$ such that:

$r{k,B}(n) = m$ for all sufficiently large $n$.

The hint is to consider $(\sum_{i} z_i)^k$ for some complex $|z| <1$.

Attempts:

$(\sum_{i} z_i)^k = b(z) + m(z^c + z^{c+1} + ...) = b(z) + m z^c / (1-z)$

Now I'm stuck.

Question: anyone have advice on what to try next?

question deleted

(exercise 1.3.10 of additive combinatorics)

Problem:

Prove that there does not exist $k, m \geq 1$, (where $k,m$ are not both 1) and $B \subset N$ such that:

$r{k,B}(n) = m$ for all sufficiently large $n$.

The hint is to consider $(\sum_{i} z_i)^k$ for some complex $|z| <1$.

Attempts:

$(\sum_{i} z_i)^k = b(z) + m(z^c + z^{c+1} + ...) = b(z) + m z^c / (1-z)$

Now I'm stuck.

Question: anyone have advice on what to try next?