Let $x \in R^n$ be vector with entries $x_1\geq\cdots \geq x_n$.
I would like to decompose this vector into vectors $x^1=(x_1,\ldots,x_m,0,\ldots0), x^2=(0,\ldots,0,x_{m+1},\ldots,x_1,0,\ldots,0)$..., where $2x_m>{x_1}$; $2x_{m+1}<{x_1}$; $x_{m+1}<2x_l;x_{l+1}>2x_{l+1} $ and so on.
How many of those vectors $x^i$? (My filing it should be $log$ terms, but I would like to see the proof of it. Any good sours will be also helpful).
Thank you.