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I'm trying to prove that for arbitrary $c > 0$,

$f(n) = o(n^c) \rightarrow \exists \epsilon > 0 \ s.t. f(n) = O(n^{c-\epsilon})$

Intuitively, this seems to be true to me (little-o implies some gap "between" $f(n)$ and $n^c$ that we can formally express as $\epsilon$). However, I'm having some trouble formalising this argument.

What I've done so far:

$f(n) = o(n^c) \rightarrow \exists c_0, n_0 > 0 \ s.t. \forall n \geq n_0, 0 \leq f(n) < c_0 n^c$

Suppose (for a contradiction) that no such $\epsilon$ fulfilling the above condition exists.

Then we have

$\forall \epsilon > 0, \exists n \geq n_0 \ s.t. f(n) \geq c_0 n^{c-\epsilon}$

I'm not sure how to proceed from here.

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  • $\begingroup$ what does s.t. stand for $\endgroup$ – shi95 May 19 at 8:29
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This is false. Consider for example $$ f(n) = \frac{n^c}{\log n}. $$

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  • $\begingroup$ Thank you for your counterexample! If I may ask, how did you come up with this counterexample? (Your thought process, etc.) $\endgroup$ – John Doe Jan 28 at 8:39
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    $\begingroup$ @JohnDoe $\log n$ is a well-known example of a "slowly" diverging function which is $O(n^\epsilon)$ for any $\epsilon>0$. Note that $\log$ is a well-known function, so it helps knowing its properties, including its asymptotic behavior. Once one knows this, it's easy to adapt it to your case. $\endgroup$ – chi Jan 28 at 9:38
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Yuval's counterexample $f(n)=\dfrac{n^c}{\log n}$ shows that there may not exist $\epsilon > 0$ such that $f(n) = O(n^{c-\epsilon})$ although $f(n) = o(n^c)$.

In fact, there is a general proposition.


(Gap between $o$ and $O$). Let $f_\epsilon(n)$ be a nondecreasing positive function on $\Bbb N$ parameterized by $\epsilon\ge0$ such that $f_{\epsilon_1}(n)\ge f_{\epsilon_2}(n)$ and $f_{\epsilon_2}(n)=o(f_{\epsilon_1}(n))$ if $\epsilon_1<\epsilon_2$. In plain words, $f_\epsilon(n)$ becomes bigger when $\epsilon$ becomes smaller. Then there exists a function $f(n)$ such that $f(n)=o(f_0(n))$ and $f(n)\not=O(f_\beta(n))$ for any $\beta>0$.

Proof. Let $k>0$ be an integer. Since $f_{\frac1k}(n)=o(f_{\frac1{k+1}}(n))$ and $f_{\frac1k}(n)=o(f_0(n))$, there exists constant $c_k>0$ such that $f_{\frac1{k}}(n)\le \frac1k(f_{\frac1{k+1}}(n))$ and $f_{\frac1{k}}(n)\le \frac1k(f_0(n))$ if $n>c_k$. We can assume $c_k\lt c_{k+1}$; otherwise, we can replace $c_{k+1}$ by $\max(c_k+1, c_{k+1})$ recursively.

Let $f(n)$ be defined as the following. $$f(n) = f_{\frac1k}(n) \text{ when }c_k\le n\lt c_{k+1}.$$

We can verify that $f(n)=o(f_0(n))$ and $f(n)\not=O(f_\epsilon(n))$ for any $\epsilon>0$.


Here are a few related exercises.

Exercise 1. Show that $\log n=o(n^\epsilon)$ for any $\epsilon >0$.

Exercise 2. Show that $n=o(e^{\epsilon n})$ for any $\epsilon >0$.

Exercise 3. Show that $n=o(e^{\epsilon n^c})$ for any $\epsilon >0$ and $c\gt 0$.

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