I have difficulties in understanding the notion of density for distribution.
Notion of density for distribution. A distribution $H$ over $\{0,1\}^n$ has density $\sigma$ if for every $x \in \{0,1\}^{n}$, $Pr[H=x] \leq \frac{1}{2^n\sigma}$.
The following is my desperate tries to understand the notion. if $H$ is uniform distribution over $\{0,1\}^n$ then for every $x \in \{0,1\}^n$, $Pr[H=x]=\frac{1}{2^n}$.
if a distribution $H$ has density $\sigma = 1$ then $P[H=x]\leq\frac{1}{2^n}$, so distribution $H$ is upper bounded by the uniform distribution.
if a distribution $H$ has density $\sigma = \frac{1}{2}$ then $P[H=x]\leq\frac{1}{2^{n-1}}$, so distribution $H$ is upper bounded by the uniform distribution over $\{0,1\}^{n-1}$.
if a a distribution $H$ has density $\sigma = \frac{1}{2^n}$ then $P[H=x]\leq 1$.
So density $\sigma$ might determine the part of distribution over $2^n$ where the actual "probabilistic weight" should be placed? However it's always upper bounded, therefore we can always say something about the upper bound?
As you see I don't have intuition behind this notion and I would appreciate any help.