Functions less that $N-N^\theta$ for all $\theta<1$

Question

I am working with a supervisor on a topic about normal numbers, and a certain class of functions came up. It is the class for which $0 \le f(N) \le N-N^\theta$ for all $0<\theta<1$. He said that there are many such functions, but I can't understand how there can be other than the zero function. I was wondering if anyone can help explain how there can be a function that fits this and what it might be? Any help is appreciated!

Answer 1

The problem is when $\theta$ is close to $1$, so let $\theta = 1-c$ where $c$ is small.

$N-N^{\theta} =N-N^{1-c} =N(1-N^{-c}) =N(1-\dfrac1{N^c}) \gt N(1-\dfrac1{\ln N}) $ since $\ln N < N^c$ for any $c > 0$ for all large enough $N$.

Therefore one possibility is $N-N/\ln N$.