TITLE: Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
QUESTION [19 upvotes]: For every nonnegative integer $n$ and every real number $ x$ prove the inequality:
$$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$

REPLY [6 votes]: We can use the following inequality:

$$|\cos x|\leq\tfrac12 \implies |\cos x|+|\cos 2x|\geq1.$$

Proof: use symmetry to reduce to the case $\cos x\in[\pi/3,\pi/2]$ where $\cos x-\cos 2x-1=(1-2\cos x)\cos x\geq 0.$
We can amortize the terms with $|\cos 2^kx|\leq \tfrac12$ by combining them with the next term, so every term effectively counts for $1/2$ except possibly the last. This gives the bound
$$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq\frac n2.$$