Entire functions and discrepancy

Letf be an entire function that is real on the real axis and not a polynomial. Let 1<α<4/3. A condition $$\log \left| {f(z)} \right| = O((\log \left| z \right|)^a )$$ is known to guarantee uniform distribution of the sequencef(n) (n=1,2...). However, we show here by an example that no quantitative version of the uniform distribution can be deduced from (1).