A conjecture on bounded complex partial sums A friend of mine has made the following conjecture, but we don't know how to prove it.

Let $(a_n)_{n\in \mathbb {N}}$ be a strictly increasing sequence of natural numbers. 
Suppose that for every complex $z\neq 1 $ with $|z|=1$, there exists a constant $C_z&gt;0 $ verifying
$$\forall N\geq 0, \qquad \left|\sum_{i=0}^N z^{a_i}\right|  &lt;C_z$$
Then the sequence $(a_n) $ contains all but finitely many natural numbers.

For exemple, if $a_n=2n$, the property is not verified with $z=-1$.

Can you find a proof or a counterexample?
I have failed to prove this conjecture and am currently trying to find a counterexample.
Any linear transformations (eg $$a_n = k*n + r$$)  won't work as a counterexample since unit roots will be a problem so I've tried looking at sequences like $$a_n = n^2$$ 
I have not proven that the series converges (it doesn't seem easy since usual tests won't work) but according to wolfram alpha it does converge even if |z| = 1. Which would mean that the conjecture is false.

http://m.wolframalpha.com/input/?i=sum+of+z%5E%28n%5E2%29+from+0+to+infinity

I don't really know what the strange looking function is though (when I look at the latex code it's written "curly theta") and it might be above my level to show that the series converge if |z| =1.
I have failed to prove this conjecture and am currently trying to find a counterexample.
Any linear transformations (eg $$a_n = k*n + r$$)  won't work as a counterexample since unit roots will be a problem so I've tried looking at sequences like $$a_n = n^2$$ 
I have not proven that the series converges (it doesn't seem easy since usual tests won't work) but according to wolfram alpha it does converge even if |z| = 1. Which would mean that the conjecture is false.

http://m.wolframalpha.com/input/?i=sum+of+z%5E%28n%5E2%29+from+0+to+infinity

I don't really know what the strange looking function is though (when I look at the latex code it's written "curly theta") and it might be above my level to show that the series converge if |z| =1.